After England's record 68-14 rout of Wales earlier in the day in Cardiff, Les Bleus needed to beat the Scots and they did so with a four-try, bonus-point victory.
Ramos' haul included a try as part of a 19-point blitz in 20 minutes just after the interval which sealed France's second championship in four years, but they missed out on the Grand Slam after their round two loss at England.
"What we wanted to do tonight was to lift the trophy and it's job done," France captain Gregory Alldritt told France Television.
"We've had painful experiences in the past, and these moments you have to make the most of them," he added after France finished second four times in six editions of the tournament.
Scotland finished fourth in the table after a third loss in this year's tournament despite some encouraging displays from their British and Irish Lions hopefuls, including full-back Blair Kinghorn and fly-half Finn Russell.
France coach Fabien Galthie was forced into two changes from last weekend's huge victory over holders Ireland as scrum-half Maxime Lucu and centre Gael Fickou came in for the injured Antoine Dupont and Pierre-Louis Barassi.
Scotland's Gregor Townsend also made a couple of alterations with lock Gregor Brown making his first Six Nations debut and Matt Fagerson in at No 8 from last Saturday’s nervy win over Wales.
The biggest cheer from the expectant crowd in the opening 10 minutes was when France talisman Dupont appeared on the stadium’s big screen, a week on from suffering a serious knee injury against Ireland.
France controlled the opening quarter to lead 10-0 thanks to a Moefana try sandwiched between a Ramos penalty and conversion, with the seven-pointer coming with Scotland flanker Jamie Ritchie in the sin bin.
By the 25-minute mark, the hosts were down a man themselves as hooker Peato Mauvaka was yellow carded for a headbutt on scrum-half Ben White, tipped as a member of Scotland’s talented backline to be part of this summer’s Lions squad touring Australia.
Record-breaker Ramos
Russell cut the deficit before Ramos reinstated the 10-point advantage to make it 13-3, which moved him ahead of Frederic Michalak on 436 points as Les Bleus' all-time leading points scorer.
With Mauvaka off, Russell found his groove with the visitors on the up, to set up winger Darcy Graham.
The maverick playmaker’s conversion made it 13-10 with half an hour played.
Mauvaka returned only to cross paths with prop Jean-Baptiste Gros, who was sent to the bin before Russell brought the sides level with a penalty.
Ramos re-established the three-point lead three minutes from the interval but Scotland ended the half the better and were denied the lead, as Kinghorn’s foot was adjudged to be in touch in the build-up to Tom Jordan’s try.
Just after the break, France put Scotland’s hopes to bed as home winger Louis Bielle-Biarrey scored a record-equalling eighth try in one campaign of the Six or Five Nations, benefitting from a Graham knock-on.
Ramos added the extras to make it a 10-point game with 35 minutes left before Galthie introduced five from an unusual seven forwards on the bench.
The changes among the pack had the desired effect as a dominant rolling maul led to Ramos strolling over just before the hour mark.
His sixth successful kick at goal made it 30-16, with no way back for Townsend's tiring team, and sent the jubilant home fans in full voice with a mid-game rendition of La Marseillaise.
A second try from Moefana, again from a Fickou pass in the final quarter made sure of victory and the title, a seventh Six Nations championship success for France.
England score 10 tries in Wales thrashing
England had their match all but wrapped up at half-time after scoring five tries to lead 33-7 at the break.
England captain Maro Itoje opened the scoring as early as the third minute, with Tom Roebuck, Tommy Freeman, Chandler Cunningham-South and Will Stuart all following the skipper's lead.
Scrum-half Alex Mitchell, 20-year-old debutant replacement Henry Pollock, with two of his own, Joe Heyes and Cunningham-South again added five more tries between them in the second half.
That allowed England to surpass their previous record winning Six Nations winning margin over Wales of 40 points set during a 50-10 success at Twickenham in 2006.
Freeman's effort meant he equalled France wing Philippe Bernat-Salles' 2001 record of scoring a try in every round of the Six Nations.
For Wales, their 17th successive Test defeat represented an unwanted record for a Tier One country in the professional era. Their 11th straight championship loss meant they had finished with back-to-back wooden spoons for the first time in the history of a proud rugby nation.
It was not the way interim Wales coach Matt Sherratt would have wanted to wrap up his three-game caretaker spell after taking over mid-tournament following the end of Warren Gatland's unsuccessful second stint in charge.
Itoje had spoken in the build-up about the need to "start well" and "really take the game to Wales". The lock forward was as good as his word under the closed roof of a raucous Principality Stadium.
England kicked a penalty to the corner and, following a couple of line-out drives, Itoje dived over from a ruck as Wales were caught napping. Fly-half Fin Smith converted and England led 7-0.
Wales full-back Blair Murray sprinted in five minutes later but his try was disallowed after scrum-half Tomos Williams was offside in the build-up.
But there was no stopping a powerful England and Roebuck marked his first Test start with a 10th-minute try when the wing shook off the diminutive Murray to go in at the right corner. Fin Smith added the difficult conversion and England were 14-0 ahead.
Murray, however, almost broke clear for a try after chipping ahead only for England hooker Luke Cowan-Dickie to bring him down with a brilliant tap-tackle.
Wales did cut the deficit in the 31st minute when, following an attacking line-out off a penalty, Williams' pass sent in centre Ben Thomas for a try converted by Gareth Anscombe.
Just three minutes later, England pulled clear again. Fly-half Anscombe's clearing kick was charged down and England worked out the ball back to Fin Smith before Freeman went over
England, in arguably the slickest 40 minutes of coach Steve Borthwick's reign, still had time to score two more tries before the interval. Wales were once more out-muscled at the breakdown with Cunningham-South, on as a replacement for the injured Ollie Chessum, twice involved before he finished for the four-try bonus point.
England, with flanker Ben Curry outstanding, crossed Wales' line again as prop Will Stuart celebrated his 50th cap with only his third Test try. Fin Smith then missed his first conversion in five attempts - the only blemish in a dominant first-half display by England.
Wales rallied at the start of the second half but it was England who scored next when Mitchell, seizing on a loose ball, kicked ahead for the visitors' sixth try.
Pollock and Heyes cemented England's grip on the game before Thomas scored his second try of the match only for Cunningham-South to have the last word late on.
Sheehan stars as Ireland edge past Italy
Earlier in the day, Dan Sheehan scored a hat-trick of tries as the Irish got the bonus-point victory they needed on the day, Peter O'Mahony and Conor Murray said goodbye to international rugby in front of an army of travelling fans.
Ireland had to come through a tough clash with battling Italy who played a huge chunk of the match with fewer men on the field due to Michele Lamaro and Giacomo Nicotera both being yellow carded and Ross Vintcent's red card for a reckless tackle on Ireland's other try scorer Hugo Keenan.
Italy were again undone by the ill-discipline which has so often cost them in the Six Nations and could have left them holding the wooden spoon even though they collected a defensive bonus point.
Gonzalo Quesada's side remained fifth on five points, two points above bottom side Wales, who would have avoided finishing last with a victory over England or if collected enough bonus points while maintaining their superior points difference on the Italians.
"Obviously we're disappointed at the end of the match because it was a day in which 15 against 15 we were for long periods the better team," Quesada told reporters.
"We were not all that far away of getting a win which would have been incredible, but at the same time, there's a lot of which we can be proud."
The hosts were on top of lacklustre Ireland in the opening period but went in behind at the break in large part due to Lamaro getting his yellow card and giving away a scrum which led to Sheehan putting the away side ahead with their second try of the game.
Up to that point, Italy were leading thanks to Monty Ioane's converted score and a Tommaso Allan penalty but Vintcent's awful challenge shortly after half-time led to a classic wobble from the Italians who were kept in the game by Ireland failing to convert three of their four tries.
Finlay Bealham, James Lowe and Keenan were also denied tries for Ireland by the TMO in an incident-packed match.
Substitute Stephen Varney made sure it would be a tense finale when he touched down for a converted score after Ange Capuozzo chased down a long kick and forced an Irish handling error, but Ireland held out for the win.