The home side were determined to present McGeechan with a face-saving victory in his final game in charge - and salvage some pride from a disappointing campaign.
The Scots had managed just three tries in their opening four matches but doubled their tally by half-time at Murrayfield with touchdowns from James McLaren, Kenny Logan and Jason White.
But they were never able to fully shake off the spirited Azzurri and, even though winger Chris Paterson finished off some smart play by Simon Taylor to grab a fourth try in the second half, the result was in doubt right until the final whistle brought relief to the 45,739 crowd.
Paterson also kicked five goals from seven attempts to reach a century of points for his country while fellow winger Logan turned back the clock with a vintage performance to claim the man-of-the-match award.
Italy, who began the championship with a win over Wales, were in sight of their first away victory when full-back Mirco Bergamasco opened the scoring with a fourth-minute try.
Fly-half Ramiro Pez also scored a try, in addition to kicking four goals, and replacement forward Scott Palmer added a third try on 68 minutes to keep the visitors in touch.
However, Scotland were able to hold on for a morale-boosting victory which ended their campaign on a positive note.