The well-deserved win ends a run of nine straight Six Nations defeats for the Scots.
Skipper Greg Laidlaw kicked 21 points with tries from John Barclay, John Hardie and a late Tommy Seymour try sealing the win from a lovely Stuart Hogg pass.
Scotland lost both their opening clashes with England and Wales after letting golden opportunities slip at vital moments, but they were never in danger of tripping up this time after flying out of the traps.
Barclay was teed up by a clever offload from Hogg after 10 minutes while Hardie dived over just after the quarter hour following Finn Russell's daring break.
Ghiraldini trimmed the deficit back after 30 minutes before Fuser squeezed the ball down for Italy's second midway through the second half after Russell was sent to the sin bin.
But Scotland - thanks to Laidlaw, whose kicking kept the scoreboard ticking over - always held the hosts at arm's length.
WP Nel was also shown a yellow card with eight minutes remaining but Scotland refused to panic and eventually sealed the victory they craved so badly two minutes from time when Hogg produced another stunning assist for Seymour as he tossed the ball to his Glasgow team-mate from behind his back.