Tommaso Allan, who had represented Scotland at junior level, scored 22 points, including two tries, as the Azzurri looked set to end a 16-match losing run in the tournament.
However, having led 24-12 early in the second half at Stadio Olimpico, the hosts faded badly down the stretch to suffer yet more disappointment.
Scotland capitalised on Italy's tiredness, tries from Sean Maitland and Stuart Hogg, both converted by Laidlaw, putting it 26-24 ahead as the game moved towards a climax.
Allan slotted over a penalty to briefly put Italy back ahead, only for Laidlaw to respond with a successful kick of his own in the 79th minute that finally settled a see-saw contest in Scotland's favour.
The win means Gregor Townsend's squad finishes the campaign with three victories and two defeats, as it did in 2017.