Anthony Martial put United ahead after 10 minutes, but Naldo equalised before the Bundesliga runners-up went ahead following a delicious 11-pass move that was finished by Vierinha.
A late own goal by Josuha Guilavogui gave United hope but Naldo found the net shortly after to earn a 3-2 win and send the injury-hit three-time European champions out.
CSKA Moscow were unable to pull off a shock win over PSV Eindhoven, and United now face the prospect of playing Europa League football in the new year.
Manager Louis van Gaal wanted to guide United to Champions League glory, but he will have to get used to long trips to far-flung corners of Europe on Thursday nights instead.
United may have been beset by injuries, but Van Gaal cannot use that as an excuse
United should have cruised through this group
The Dutchman's critics will now only grow louder.
Without seven first-team regulars, Van Gaal was forced to hand defender Guillermo Varela his full debut
The Wolfsburg fans did their best to try to make David Moyes' first signing feel uncomfortable and so did the players tearing about the pitch in white and green.
The hosts lived up to their nickname of Die Wolfe and hounded United like a pack of aggressive wolves.
Van Gaal's team were never allowed to settle on the ball
Varela drifted inside, allowing Andre Schurrle a chance to shoot but he fluffed his lines and United breathed again.
Wolfsburg were well on top but somehow found themselves behind in the 10th minute.
Daley Blind found Juan Mata, who carved open the Wolfsburg back four to set Martial free
The Frenchman went into the game in poor form, but his finish was exemplary
He took one touch before rolling the ball past Diego Benaglio.
It was a superb finish, but Wolfsburg equalised with an even better one just three minutes later
Ricardo Rodriguez whipped in a low cross and Naldo got in front of Chris Smalling before volleying past David de Gea
United were back to square one.
Fellaini almost put them ahead again with a close-range header but Benaglio saved
Wolfsburg went on the attack at a devastating cost for United.
Wolfsburg switched the play from right to left and Schurrle drifted infield before laying the ball off to the sensational Julian Draxler
He glided past Bastian Schweinsteiger and two more United players while playing a one-two with Max Kruse before squaring to Vierinha and he slotted the ball in from close range
It was a stunning goal, one worthy of a team seemingly bound for a place among Europe's elite.
Draxler drew yet more gasps from the despairing United fans when he wriggled free in the box, but De Gea saved his shot.
United were reeling
They needed the half-time whistle to regroup
Instead they were handed another problem as Darmian had to be replaced by 19-year-old Cameron Borthwick-Jackson because of an injury to the Italian.
United thought they had equalised when Jesse Lingard's ball sailed into the goal, but the referee ruled Mata, who was in an offside position, had interfered with play.
United performed better at the start of the second period
Memphis Depay drew a save from Benaglio with an acrobatic volley
Wolfsburg still looked dangerous though
De Gea tipped an effort over from Schurrle and Kruse missed a sitter.
Van Gaal threw on Nick Powell for his first game in almost a year.
United supporters' spirits were raised when CSKA took the lead, but PSV soon equalised and Van Gaal's men were heading out
They needed an equaliser and they got it when Guilavogui knocked Fellaini's header into his own net, but just a minute later Naldo headed in to send the locals inside the Volkswagen Arena wild.
United pushed for an equaliser but it did not come
It would not have mattered anyway
PSV's winner in Eindhoven meant United were already out.
TWEET OF THE MATCH
" #VanGaal under pressure tonight...." - German tennis great Boris Becker adds to that pressure on the United boss.
PLAYER RATINGS
Manchester United -
David de Gea: 8 (out of 10)
Matteo Darmian: 4
Chris Smalling: 5
Daley Blind: 5
Guillermo Varela: 6
Marouane Fellaini: 6
Bastian Schweinsteiger: 4
Jesse Lingard: 5
Juan Mata: 5
Memphis Depay: 5
Anthony Martial: 6
Subs -
Cameron Borthwick-Jackson (for Darmian 43): 5
Michael Carrick (for Schweinsteiger, 69): 5
Nick Powell (for Mata, 69): 5
STAR PLAYER
David de Gea made several outstanding saves to keep Wolfsburg down to three goals
This is his level, but too many team-mates were off their game and looked out of place.
MOMENT OF THE MATCH
Wolfsburg's second goal was a thing of great beauty, the German side cutting through United's defence with incisive passing that bewildered the Red Devils backline
United used to play like that.
VIEW FROM THE BENCH
Louis van Gaal must wish he had Wayne Rooney available for the game
The United skipper often raises his game on such occasions and his presence was missed.
MOAN OF THE MATCH
United's defence was shambolic for Wolfsburg's first goal, and although the volleyed finish by Naldo was a special strike, the free-kick into the penalty area should have been cut out before it found its way through.
WHO'S UP NEXT
Bournemouth v Manchester United (Barclays Premier League, Saturday, December 12)
Source : PA
Source: PA