Centre Alex Dunbar scored two second-half tries for Scotland who fought back from 13-3 down at the interval to secure their first win of the tournament and relieve the pressure on under-fire coach Scott Johnson.
Italy's Tommaso Allan contributed 13 points in the first half and Joshua Furno crossed for a try 10 minutes from time to put the Italians 20-18 ahead.
But the Scotland forwards set up a solid attacking position and with seconds remaining Weir slotted over a drop goal to send the thousands of Scottish fans in Rome wild.
(Reporting by Terry Daley, editing by Ed Osmond)
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