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 by Naldo`s late header.
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 the Red Devils` result.
CSKA led 1-0 in Eindhoven as United trailed, before both PSV and United were briefly level in their respective games.
But late goals 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.
Manchester United eliminated from the Champions League
09 December 2015
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The Red Devils made an ideal start when Anthony Martial caressed Juan Mata`s pinpoint pass into the bottom corner.
