Allan's third penalty put Italy ahead but Pyrgos went over against the run of play late on and Weir added the conversion to seal the Scottish win.
The two sides meet again at Murrayfield next Saturday before Scotland coach Vern Cotter announces his 31-man squad for the World Cup.
Scotland begin their World Cup campaign against Japan in Gloucester on Sept. 23.
(Reporting by Ed Osmond; Editing by Ken Ferris)
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