Chris Hussey pounced twice in the second half to stun the home crowd and ensure it was the League Two club, not David Weir's side, which reached the second round.
United captain Michael Doyle had earlier threatened to take the tie into extra time when he levelled from long range but the Albion defender had other ideas.
Rowett said: "It goes without saying that I'm delighted to go through.
"Those were two great finishes from Chris but we also had to show the other side of our character at times because United are a good team.
"They put us under pressure for periods. But we dug in and took our chances when they arose."
Rowett added: "We always knew we'd have to work with and without the football. The lads did exactly what we asked of them and I can't praise them enough.
"The challenge now is to try and ensure we maintain the same standards. Especially in the league."
Blades boss Weir was naturally disappointed to exit the competition but also tried to take some plus points from the performance.
"There were positives and also, of course, some negatives," Weir said. "Obviously I'm not happy that we didn't go through.
"I thought we played well enough but there are things that we still have to work on. We saw some of those things out there.
"We know that ultimately we'll be judged on our ability to win football matches and we haven't done that here.
"We want to pass the ball, we want good movement and we want to play at a high tempo.
"But I come back to the fact we want to do that within a winning scenario."
Weir added: "We've got to look at why we conceded the goals we did. Discover why we did and work on ensuring it doesn't happen again.
"Like I say, it wasn't all bad. But obviously there is plenty we can do better."