The 26-year-old Edinburgh wing missed the Six Nations with a knee injury but, after scoring a hat-trick in his last match for the Scots against Argentina in November, he picked up where he left off.
Replacement Josh Bayliss sealed the victory with the Scots' third try in the last play of the match after Monty Ioane's score just after the hour had given the Italians hope.
Graham was one of the few established internationals in Gregor Townsend's starting XV as the Scotland coach rested key players ahead of a double-header against World Cup hosts France.
The visitors started well, but Scotland made the breakthrough in the 13th minute when Graham, on the right wing, collected Ben Healey's kick and helped himself to a straightforward touchdown.
The Azzurri edged in front by half-time courtesy of two penalties from Harlequins stand-off Tommaso Allan.
Scotland were back in front eight minutes after the interval when Healy scored a penalty. In the 56th minute, Graham claimed his second try of the match when he received a reverse pass from Ali Price and bounded over the line.
Healy was accurate with the conversion as the Scots opened up a 15-6 advantage.
Italy got back into the match just after the hour, when Ioane went over on the left after a lovely pass from Allan, who converted.
The hosts regained control when Healy kicked another penalty in the 74th minute. They finished on a high note, with substitute Bayliss scoring his first international try, converted by the impressive Healy.
"It was a bit of a mixed bag," Scotland captain Rory Darge told Amazon Prime. "Loads of good stuff, especially in defence.
"The shots and physicality was really good but we'll have a lot to work on come Monday, especially taking care of the ball. It was tough conditions but we could have done a bit more, especially around the breakdown."
