The 26-year-old collided with Black Cats midfielder Wahbi Khazri.
"It is dislocated so he is out for several weeks," said United boss Louis van Gaal after Saturday's game.
Luke Shaw, Marcos Rojo, Ashley Young, Antonio Valencia, Phil Jones and Guillermo Varela are out - leaving Cameron Borthwick-Jackson and Donald Love as the club's only fit full-backs.
Love, 21, who was on loan at Wigan earlier this season, came off the bench during the Premier League game to make his first-team debut. Saturday was also the first time he had featured in United's first-team squad.
Borthwick-Jackson, 19, made his first-team debut in November.
"We don't have any more full-backs in this club," added Van Gaal. "I have to line up a 13-year-old guy or someone like that."
Darmian joined United from Italian side Torino last summer and started at right-back against Sunderland.

Ezewele, 20, joins on a one-year deal with the option for a further 12 months and is Fylde's sixth summer signing.
He started his career at West Brom but failed to make a first-team appearance before moving to Yeovil in July 2016.
The defender will link up again with Kidderminster team-mates Jordan Tunnicliffe and Zaine Francis-Angol, who have also joined the Coasters.
Find all the latest football transfers on our dedicated page.
Matteo Darmian added to Manchester United's full-back problems after dislocating his shoulder at Sunderland.
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Newly promoted National League side AFC Fylde have signed full-back Josh Ezewele from Kidderminster Harriers.