Jack Ross' side were worthy of the victory, with United struggling to create many opportunities against a resolute Black Cats backline.
Blades boss Chris Wilder made 10 changes to the side that won 2-0 at Everton on Saturday, with only Callum Robinson keeping his place in the starting XI, while summer signings Ravel Morrison, Lys Mousset and Phil Jagielka were handed starts.
Sunderland also rotated their squad, showing eight changes, with summer signings Joel Lynch and Laurens de Bock making their debuts.
Robinson had the first chance of the game when his curling effort was well saved by Sunderland goalkeeper Lee Burge.
Power, one of those Sunderland players recalled to the starting line-up, then responded with a thunderous opener after just nine minutes.
After a corner was cleared into his path, the former Wigan man took a touch before unleashing a fearsome drive into the top corner from 25 yards.
United came close to an equaliser when Robinson broke clear down the left, but his low cross was just ahead of the onrushing Mousset.
At the other end, Elliot Embleton drew a smart stop from Blades goalkeeper Simon Moore as the League One side pushed for a second.
The Black Cats went even closer at the start of the second half when Luke O'Nien's long-range effort clipped the outside of the post.
United pushed forward in the closing stages in search of an equaliser but they couldn't find a way past the impressive Burge, who saved well from Ben Osborn late on to secure Sunderland's place in the next round.

Carabao Cup draw for the fourth round:

Chelsea play Manchester United, Liverpool host Arsena while Colchester travel to Crawley in fourth round of the Carabao Cup.
Ties played w/c 28 October

Everton v Watford
Aston Villa v Wolves
Man City v Southampton 
Buron v Leciester 
Crawley v Colchester 
Chelsea v Man Utd
Oxford v Sunderland
Liverpool v Arsenal