Wawrinka endured a first-set ordeal against an opponent 17 years his junior, winning only seven points on his own serve as world number 249 Musetti converted three of four break points on offer.
The 2015 French Open champion then dropped serve once more in the opening game of the second set before finally getting on the board at 1-2 and breaking back to level things up at 3-3.
The 35-year-old was able to battle on to a tie-break but Musetti reeled off five straight points from 1-1 to pull clear decisively.
US Open quarter-finalists Andrey Rublev and Denis Shapovalov, seeded ninth and 12th in Rome respectively, eased to straight-sets wins.
Rublev's debut in the competition yielded a 6-4 6-4 triumph over Facundo Bagnis to set up a second-round meeting with Hubert Hurkacz while Shapovalov dazzled with 23 winners and 12 of 15 points at the net as he dominated Guido Pella 6-2 6-3.
Pedro Martinez is up next for the Canadian after he upset Sam Querrey, while another qualifier, Dominik Koepfer saved match point to beat Alex de Minaur 3-6 6-3 7-6 (8-6). Koepfer will meet fifth seed Gael Monfils next.
Novak Djokovic begins his campaign against Salvatore Caruso, the Italian wild card who similarly prevailed in a third-set tiebreak to down Tennys Sandgren.
Milos Raonic, defeated by Djokovic in the Western & Southern final and seeded 13th, beat Adrian Mannarino 7-6 (7-3) 6-2.