The Sky Bet League One team, finalists in this competition as recently as 2014, led courtesy of Power's ninth-minute effort and it was them who came closest to scoring the game's second goal when Luke O'Nien's shot struck a post in the second half.
The opening seven minutes aside, the Blades offered very little and few could deny Sunderland their victory.
The hosts made 10 changes. Only Callum Robinson survived from the team who started the Premier League win at Everton on Saturday. He led the attack alongside Lys Mousset, who was making his full debut for the Blades.
In the early stages, Chris Wilder's side looked far more fluent than they did against Blackburn in the previous round.
Robinson brought a good parrying save from Lee Burge before the Sunderland goalkeeper was called into action again to smother a Luke Freeman shot. Burge also had to be quick off his line to save at the feet of Robinson.
However, despite the Blades' sharp start, it was Sunderland who struck first after nine minutes. After the hosts could only clear a corner to the edge of their penalty area, Power found the top corner of the net with a 25-yard strike.
Sunderland were playing with confidence and had managed to stem the flow of home attacks and as the first half progressed, the Premier League team lacked ideas and had lost their way.
The visitors had not threatened much either since the goal but came close to extending their lead early in the second half when O'Nien's shot from six yards out struck the outside of a post.
However, little changed for the Blades and midway through the second half John Fleck was introduced for Kean Bryan.
Still the hosts struggled to create chances. Clarke dragged a shot wide from inside the penalty area late on and that was as close as they came to drawing level.