The match was eagerly anticipated as a re-run of the thrilling Champions League final when two dramatic goals right at the end of the game took the trophy to Old Trafford in a devastating blow to the Germans.
This time round, United struck right at the beginning of the match, with England striker Wayne Rooney latching on to a deflected free-kick and smashing home a close-range volley for his 34th goal of the season with only 66 seconds played.
Bayern were left to rue a golden opportunity half way through the first period as Olić somehow missed from about a yard out, unmarked and in front of goal.
Rooney missed a chance of his own in front of goal before the end of an open and entertaining first half which produced only one goal but could have seen a half-dozen as both teams concentrated on attacking.
United manager Sir Alex Ferguson called on his side to step up their effort in the second half but it was the home side that started hot with several shots on goal in the first few minutes of the second period.
Bayern continued to dominate the second half, fully deserving their equaliser when a long-range free kick from French international Franck Ribery deflected off Rooney past helpless goalkeeper van der Sar.
During a frantic last 10 minutes, United defender Nemanja Vidic clattered the bar and Gomez forced van der Sar into a last-ditch save during injury time.
But van der Sar was helpless as Olić weaved his way through the United defence and slotted home in the second minute of injury time.
Ferguson accepted that the better side had prevailed.
"We didn't play well and we kept on giving the ball away," said the Scotsman. "Bayern were better than us and we can't deny that either. They pressed us really hard and we cracked a bit. The last goal was a terrible one to give away.”
As a further blow to Ferguson and England, Rooney hobbled off the pitch at full time after a clash with Gomez.
"He's got a kick in the ankle, we'll just have to see tomorrow. Let's hope it's not too serious," said Ferguson.
Bayern coach Louis van Gaal was delighted with what he described as a "well-deserved win."
After the shocking first goal, "we needed 10 minutes to recover and afterwards we were the best side. We have a good chance in Manchester because we can always score away from home."
The English side came into the clash in stunning form, having won their last seven games, including a 4-0 thumping of Italian giants AC Milan and a satisfying 2-1 defeat of fierce rivals Liverpool at home.
Coming in to the quarter-final clash, the Reds were undefeated in their last 17 European encounters, a point made forcefully by coach Sir Alex Ferguson before the game.
In contrast, Bayern's recent form has been poor, with rare back-to-back league defeats knocking them off the top of the Bundesliga.
The four-time European champions have also been less than convincing in Europe, needing a hefty dose of luck to squeeze past Italian outfit Fiorentina.