Blackburn boss Paul Ince will be hugely relieved to see his side in the quarter-finals of the cup and get one over on his former team-mate Roy Keane in the process.
Ince's much-changed side put in a less than inspiring display, but they scored from their two shots on target in the game to end a run of six matches without a victory.
Santa Cruz put them ahead within minutes of coming off the bench when he profited from a Samba flick from Martin Olsson's left-wing cross.
Martin Fulop was helpless to stop Santa Cruz's close-range effort but the Sunderland keeper was involved in the second goal when he pushed Derbyshire's cross on to the unwitting Bardsley, who could only look on in horror as the ball ended up in the net.
Within two minutes Jones, who was surprisingly named in the Sunderland side after earlier being ruled out of the game, drilled a left-footed shot past Robinson to give the home side hope.
But, having come from behind to beat Nottingham Forest and Northampton in the previous rounds, they were unable to find an escape route this time.
Sunderland manager Roy Keane: "It's like groundhog day - the same scenario, the same post-match interview - it's getting frustrating.
"There has not been much between the teams in our recent games but we're on the losing end, so we'll have to sort that out.
"We don't score enough goals. We're always putting ourselves under pressure and you know that if you don't take your chances you will get punished."
Blackburn manager Paul Ince: "We hadn't won for a few games so it is important to get a win under our belts.
"When you are 2-0 up you think the game is over and then we let one in and we have to learn from that. You can't afford to do that.
"You see how much the win meant to the lads. It was important we got back to winning ways, we have played some good football and we have deserved a bit more at times. We will enjoy it but you don't get three points and the most important thing for me is Saturday."
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