Boro made a mockery of their record of only one goal in nine away games with central defender Chris Riggott scoring twice in a seven-minute spell midway through the first half.
Kevin Phillips threw Sunderland a life-line with a superb solo effort in the 56th minute only for Kevin Kilbane to gift wrap the points for Boro with a dreadful back pass which allowed Malcolm Christie to restore their two-goal lead.
Boro manager Steve McClaren believes the Black Cats have the quality to avoid the drop, but there will have to be a dramatic improvement to Howard Wilkinson's record since replacing Peter Reid last October.
Sunderland have won just two out of 19 league matches under his reign and have taken only one point out of a possible 27 since mid-December.
Said McClaren: "Their destiny is in their own hands, there's ten games to go so who knows anything can happen. The away win has been a while coming and to score three goals was a bonus.
"I said we needed to bring players in and there is a buzz and freshness about the place now. We got a break but ran out comfortable winners.''Boro's first-half goals stemmed from left-wing corners taken by the impressive Geremi.
Goalkeeper Thomas Sorensen saved from Christie after 20 minutes only for Riggott to slam home the loose ball.
Seven minutes later Gareth Southgate's powerful drive came back into play off the upright and the England Under-21 international was again well placed to beat Sorensen.
Wilkinson was disappointed that the first two goals conceded came from set pieces and the killer third goal just when his side had forced their way back into the game.
"We have worked very hard this week defending corners,'' he said. "I think the first one was mis-hit and the third goal came just after Kevin's fantastic strike.
"I was not just thinking about a draw but a victory was possible. We are doing everything possible to rectify our situation at the moment and the Sunderland fans need to be assured of that.''