Franck Kessie's miss for Milan signalled the end of an encounter which entertained from start to finish and comfortably produced United's best display of their pre-season tour so far.
The Red Devils were lacklustre in drawing 0-0 with San Jose Earthquakes on Sunday but there was much more vigour about this performance from the first whistle.
McTominay won a flick-on to Juan Mata, who skilfully turned a volleyed pass around the corner for Sanchez to bend the ball around Donnarumma with a composed finish.
Sanchez appeared to have rediscovered the confidence of his Arsenal peak and tried to score directly from a corner on 21 minutes when Donnarumma wandered off his line, the ball landing on the roof of the net.
It was not the only time in the first half that Milan's goalkeeper was nearly caught out, with Mata's audacious lob from a quickly-taken free kick forcing the 19-year-old to backpedal and tip the ball over his own crossbar.
However, Donnarumma did have his moments including an important save from Ander Herrera, after Sanchez had burst down the left flank and pulled the ball back for the midfielder arriving late into the area.
Jose Mourinho's side started the second period in the same lively fashion as they did the first, with Sanchez volleying wide from Matteo Darmian's cross and McTominay shooting straight at Donnarumma at the end of a driving run.
Borini had a chance himself moments later when he blazed off target, after midfielder Jose Mauri had skipped past McTominay and spotted his out-to-in run.
United were also pushing hard for a second goal and Milan's substitute stopper Pepe Reina was quickly called into action on 69 minutes when Sanchez teed up Mata for a left-footed shot which was bound for the bottom corner.
The closest that either side came to claiming victory inside 90 minutes was when Borini struck the base of the post late on, and it was to be another 20 minutes until a winner was crowned in front of a gripped crowd at Los Angeles' StubHub Center.
Borini's penalty miss opened the door for United but three successive Reina saves from McTominay, Joel Pereira and Eric Bailly meant that it was Milan who passed up the chance to win the shootout through Alessio Romagnoli, who hit the post.
United's own substitute goalkeeper Pereira made three saves to make up for his own miss, which would have sealed victory for the Red Devils before sudden death.
Young midfielder Ethan Hamilton missed United's 10th penalty but was spared a debut to forget when Pereira guessed the right way to save from Mauri.
After Andreas Pereira and Herrera had each scored for a second time for the Red Devils, a poorly-judged chip over the crossbar from Kessie finally brought things to a conclusion and gifted United victory.
The end result was worth two points to United who have Liverpool and Real Madrid to play in the competition.
Relive how the penalty shootout unfolded with our minute-by-minute updates from California.
MAN UTD (4-3-3): Grant (J. Pereria, 46'); Darmian, Smalling, Bailly, Shaw (Fosu-Mensah, 74'); Tuanzebe, McTominay, Herrera; Mata (Hamilton, 89'), A. Pereria, Sanchez
MILAN (4-3-3): G. Donnarumma (Reina, 61'); Abate (Antonelli, 61'), Musacchio, Bonucci (Romagnoli, 46'), Calabria; Kessie, Locatelli (Mauri, 60'), Bertolacci (Borini, 46'); Suso, Calhanoglu, Cutrone (Kalinic, 75')