Scotland flankers John Barclay and John Hardie both powered over the line in the early stages to give the visitors control of the match after nine successive Six Nations losses.
Italy struck back with a fine try by hooker Leonardo Ghiraldini following a charge from Gonzalo Garcia to cut the deficit to 17-10 at halftime.
Scotland flyhalf Finn Russell was sent to the sin bin and Italy lock Marco Fuser barrelled over the line but scrumhalf Laidlaw continued to punish Italian indiscipline with his reliable boot and Tommy Seymour's late try sealed victory.
(Reporting by Ed Osmond; editing by Clare Fallon)
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