David Moyes's side took the lead through James Garner's first-half goal at the Hill Dickinson Stadium.
Hayden Hackney doubled the lead with his first goal since his close-season move from Middlesbrough.
Everton striker Thierno Barry netted in the second half and Dan Burn's reply for Newcastle came far too late to stop the Toffees ending their three-match winless run.
Chasing their first League Cup final appearance in 43 years, Everton will travel to fourth-tier minnows Fleetwood in the week commencing December 14.
The 'Cod Army' caused one of the biggest shocks in League Cup history on Tuesday when they beat Premier League champions Arsenal in a penalty shoot-out after a 1-1 draw at Highbury.
Everton have never won the League Cup, losing their two final appearances in 1977 and 1984.
Their fans do not need reminding that it is 31 years since the club won the FA Cup in what was their most recent trophy triumph.
British Prime Minister Andy Burnham, a noted Everton fan, is old enough to remember that success, and he was cheering from the stands as Moyes's men made light work of their last-16 tie.
Newcastle will have the chance for revenge when the sides meet again in the Premier League at St James' Park on Monday.
Everton dropped a pre-match bombshell, saying Jack Grealish was unavailable for selection "due to a breach of club discipline".
Grealish, beset by criticism of his behaviour off the pitch, was reported to have taken an unauthorised absence from training. But the on-loan Manchester City winger's absence was no problem for his team-mates.
Everton hit their stride
Newcastle tested Jordan Pickford early on when Joe Willock's low drive was palmed away by the Everton goalkeeper, who repelled Fabian Schar's long-range strike moments later.
Garner steadied Everton's nerves with a clinical opener in the 34th minute. Hackney teed up Garner just inside the area and the midfielder stretched to guide a fine finish into the far corner for his first goal since January.
Everton had hit their stride, and struck again on the stroke of half-time.
At the climax of a blistering Everton counter-attack, Barry hooked a cross into the six-yard box and Hackney timed his run perfectly to volley past Lukas Hornicek.
Frustrated Newcastle boss Matthias Jaissle sent on Joelinton, Anthony Elanga and Yoane Wissa at the interval.
Wissa had an immediate chance after sprinting onto Willock's pass, but he misjudged the high bounce and ended up heading straight at Pickford. It was a costly miss as Barry put the result beyond doubt in the 69th minute.
Kiernan Dewsbury-Hall provided the cross, and Barry lashed home from close range for his seventh goal this season.
Burn scored with a header from Willock's 88th-minute cross, sparking a brawl between several players, but Newcastle's brief show of defiance was too little, too late as Everton moved a step closer to Wembley.
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