Centre Tommaso Menoncello also crossed for Italy, who were dominant in the scrum and picked off several Scotland lineouts as they completed only their second win over their opponents in the last 16 meetings.
Scotland were well below par in a sluggish start to their campaign, but managed tries through number eight Jack Dempsey and scrumhalf George Horne, and stayed in the contest until the end.
The Scots were not helped by a second-half yellow card for replacement hooker George Turner, but were unable to play their usual high-tempo game in slippery conditions and behind a creaking set-piece.
Italy made a fast start and Lynagh crossed in the corner inside eight minutes, before Menoncello added a second to give the hosts a 12-0 lead.
It took a full 20 minutes for Scotland to click into gear, but when they did they advanced into the Italy 22 and Dempsey crossed from close range.
Paolo Garbisi landed a penalty to give Italy a 15-7 lead at halftime.
The conditions did not improve in the second period and it turned into an arm-wrestle among the forwards.
Garbisi extended Italy’s lead when Scotland's Ewan Ashman committed a high tackle on flanker Manuel Zuliani, which was checked by the Television Match Official but deemed not dangerous enough for a card.
Scotland were down to 14 players when replacement hooker George Turner received a yellow card for a dangerous entry into the ruck.
The visitors hung in the contest despite their retreating set-piece and scored a second try through Horne as they secured an attacking lineout and he finished in the corner.
Scotland took the ball through nearly 30 phases in the final play of the game but found making metres tough in the wet, and the ball was eventually held up, allowinA Italy to celebrate a famous win.