The Germans were joined in the last 16 by PSV Eindhoven, who took second place in Group B after beating CSKA Moscow 2-1 at home.
United took an early lead in Germany through Anthony Martial but poor defending at set-pieces allowed Brazilian defender Naldo to score a brace.
His first came three minutes after Martial’s goal and after Vieirinha had given Wolfsburg the lead, his late second, six minutes from time, cancelled out an equaliser for United from Josuha Guilavogui just 90 seconds earlier.
The English giants could still have progressed after Sergey Ignashevich gave CSKA the lead from the penalty spot in the Netherlands.
But PSV hit back through Luuk de Jong and Davy Propper to progress to the knock-out stages.
In other games, Cristiano Ronaldo smashed in four goals in Real Madrid’s 8-0 thrashing of Malmo to set a new group phase scoring record of 11, beating his own mark from two years ago and that of Luiz Adriano for Shakhtar Donetsk last season.
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