In a dramatic end to a pulsating encounter in Germany, United were set to reach the next round when an own goal by Josuha Guilavogui levelled the scores at 2-2 with eight minutes left.
But Wolfsburg's Brazilian centre back Naldo headed his second of the night, after cancelling out Anthony Martial's opener for United, to leave the visitors needing PSV Eindhoven to lose at home to CSKA Moscow in the group's other game.
For a brief moment late in the second half it looked like United would get their wish, but PSV rallied to level and then snatch a late 2-1 win that left them second in the standings on 10 points, two ahead of United and two behind Wolfsburg.
The result was an added blow for United's Dutch manager Louis van Gaal, who has had to fend off heavy criticism for his team's perceived negative approach throughout the season.
"We fought for the win, we couldn't have done more than we have done," Van Gaal told BT Sport. "I'm very disappointed. This is the league (where) we want to show our qualities and we cannot do that anymore."
United had taken the lead through Martial after an incisive break led by Juan Mata.
The Spanish playmaker was allowed space to turn in midfield and produced a superb defence-splitting pass to find the Frenchman racing through on goal. Martial took a touch, opened his body up and slotted into the corner.
The Germans counter-punched almost immediately, however.
Ricardo Rodriguez whipped a free kick into the area and Naldo swung his right boot at the ball to direct a pinpoint low volley past United keeper David De Gea.
GERMANS AHEAD
Wolfsburg went in front with the game's standout goal after 28 minutes when Portuguese Vieirinha put the finishing touch to an astonishing team move, orchestrated by Julian Draxler.
The Germany winger beat two defenders with a drop of the shoulder and a burst of pace before exchanging a quick one-two with Max Kruse and then unselfishly squaring for Vieirinha to sweep home.
United had a goal ruled out at the end of the first half when Jesse Lingard bent a curling shot into the net past keeper Diego Benaglio around the back of Mata.
The Spaniard, however, was offside and deemed to be interfering with play despite not touching the ball.
After both keepers showcased their abilities early in the second half, the match took two dramatic turns late on.
United levelled when Wolfsburg's Guilavogui deflected a tame header from Marouane Fellaini over Benaglio to put the visitors on track to qualify with PSV then drawing against CSKA.
But Naldo's late header, combined with another goal in Eindhoven, turned the scenario on its head and left United to drop into the second-tier Europa League competition.
(Writing by Toby Davis; editing by Ken Ferris)
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