The Red Devils made an ideal start when Anthony Martial caressed Juan Mata’s pinpoint pass into the bottom corner.
Wolfsburg replied within three minutes through Naldo’s sharp volley and led when Vieirinha finished a flowing move.
United thought they were through after Josuha Guilavogui’s own goal, only to be condemned to the Europa League through Naldo’s late header.
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Louis van Gaal’s side knew they would qualify for the last 16 by beating Wolfsburg, or if they matched PSV Eindhoven’s result against CSKA Moscow.
Twice inside the final 15 minutes, United thought they were progressing at the expense of the Dutch side, who needed to better United’s result to advance.
CSKA led 1-0 in Eindhoven as United also trailed, before both PSV and United were briefly level in their respective games.
But late winners for Wolfsburg and PSV, who won 2-1, knocked out Van Gaal’s team.
United’s late misery was a stark contrast to Manchester City, who saw fortune swing back in their favour as their neighbours unravelled.
Five goals were netted inside six minutes across the two games, with City coming back to win 4-2 against Borussia Monchengladbach and pip Juventus to win Group D.
That means the Blues will avoid the likes of Barcelona and Real Madrid in the last 16, while the Red Devils face the prospect of Europa League football in the New Year.
Excitement, but an exit
Van Gaal has been criticised by some Red Devils supporters for his pragmatic, defence-minded approach – United were booed off after Saturday’s 0-0 draw with West Ham at Old Trafford, the fifth time in nine matches they had failed to score.
However, United freed the shackles and produced an entertaining display in Germany, yet it was one that ultimately led to their exit from Europe’s premier club competition.
Martial settled any early United nerves by racing on to Juan Mata’s perfectly-weighted through ball, taking a touch before coolly sliding past Wolfsburg keeper Diego Benaglio.
Van Gaal’s side continued to create chances throughout the match, only to see Benaglio brilliantly stop Fellaini’s first-half header and Memphis Depay’s far-post volley after the break.
Fellaini’s height caused problems at the corner which led to Wolfsburg midfielder Guilavogui flicking into his own net, but United were undone by their own defensive deficiencies.
Defensive horror show costs United
While United found a solution to their recent goalscoring woes, a suspect defensive display resulted in their downfall.
Wolfsburg threatened United with blistering counter-attacks and slick, one-touch passing around the visiting penalty area.
But, more importantly, the Premier League side twice switched off at Wolfsburg set-pieces shortly after scoring themselves.
Injury-hit United had a makeshift defence, with young Uruguay right-back Guillermo Varela making his first start for the club and teenager Cameron Borthwick-Jackson playing for almost 50 minutes after replacing injured left-back Matteo Darmian.
It was a lack of organisation – and possibly experience – that allowed Brazilian defender Naldo to strike the late winner.
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Many observers were surprised that Van Gaal decided to start 22-year-old Varela ahead of former England international Ashley Young, while the exclusion of experienced holding midfielder Michael Carrick also raised questions.
Even more surprisingly, Young was still ignored when Darmian had to be replaced. Instead Van Gaal turned to 18-year-old Borthwick-Jackson, who was making only his second United appearance.
England midfielder Carrick was eventually introduced for Germany’s World Cup winner Bastian Schweinsteiger after 69 minutes, at the same time Mata was withdrawn for Nick Powell – the 21-year-old midfielder whose previous United appearance came in the 4-0 defeat by then-League One side MK Dons in August 2014.
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Van Gaal said afterwards that United were unfairly denied an equaliser shortly before half-time, a decision which he felt could have changed the complexion of the match and their Champions League future.
The visitors thought they had equalised when Jesse Lingard curled a 20-yard shot past home keeper Benaglio.
But their celebrations were cut short when the assistant referee raised his flag, after a long delay, for offside against Mata.
However, replays proved the official was correct with his decision. Despite not touching the ball, Mata’s movement in front of Benaglio clearly impacted the Wolfsburg keeper.