Crespo had scored the only goal of the knockout round first leg at Old Trafford a fortnight ago and sealed a 2-0 aggregate win at the San Siro in the 61st minute with a looping header from a Cafu cross.
The striker who is on loan from United's Premier League title rivals Chelsea broke the deadlock after an open closely-contested game.
"At that level of football it's only going to take a goal to decide it -- and it was a fantastic header from Crespo United manager Alex Ferguson told Britain's ITV television.
The delighted Argentine scorer added: "I'm very happy. It was a tricky match and we had to be careful."
Both sides were denied a first-half lead by the woodwork with United winger Ryan Giggs raking a low shot onto the post and Milan's Brazilian playmaker Kaka rattling the crossbar.
Milan took a grip on the game after the break though and with neither Wayne Rooney nor Cristiano Ronaldo finding any joy against the Italian defence chances were few for United.
Knowing his team had to score Ferguson opted for an attacking lineup with only Paul Scholes and captain Roy Keane playing as conventional midfielders and the formation ensured there was plenty of space for Milan's midfield.
Dutchman Clarence Seedorf exploited that in the eighth minute when he burst forward unchallenged and slipped the ball to Crespo who forced goalkeeper Tim Howard into action.
Milan had been forced to reshuffle their defence due to the absence through injury of Georgian left back Kakha Kaladze with former United centre-half Jaap Stam partnering Alessandro Nesta in the middle and Paolo Maldini moving out to the left.
Stam who has only played nine Serie A games in an injury-plagued season looked sharp and determined from the outset making a serious of strong challenges and even bursting forward on two occasions.
The plus side of United's fluid system was that it enabled them to attack in numbers and they went very close to an opener in the 28th minute.
Dutchman Ruud van Nistelrooy picked out Giggs unmarked on the left and the Welsh winger struck the outside of Dida's far post with a low drive.
Nine minutes later Milan hit the woodwork when United defender Wes Brown failed to clear a hooked cross from Brazilian Cafu whose compatriot Kaka rattled a first-time shot against the bar.
Milan took firm control after the break though with Crespo twice worrying the United defence.
Howard was out quickly to foil him in the 56th minute as Crespo sought to meet a ball from Cafu and moments later the Brazilian provided a cross that the forward header over.
That should have been enough warning for United of where Milan's main threat lay but it was the South American pair who combined to put the Italian champions ahead in the 61st minute.
Alessandro Nesta picked out Cafu in space on the right and Crespo having lost his marker met the Brazilian's deep cross with a superbly weighted and directed header that looped over the helpless Howard.
Despite their two-goal lead on aggregate Milan continued to push forward looking to put the outcome beyond doubt and they had an appeal for a penalty turned down when Roy Keane appeared to handle while falling onto the ball inside the United area.
United now needed to score twice against one of the best defences in Europe and they should have pulled level on the night in the 70th minute but Van Nistelrooy fired wide of the target from an excellent position at the back post.
That was the last real chance for United who will be disappointed not to have troubled Dida more.
Milan's Gennaro Gattuso went close to a rare goal near the end as he dribbled past two defenders and poked the ball just wide after a back-heeled return pass from Kaka but the Italians had already done enough to deservedly reach the last eight.