Louis van Gaal's United made a promising start at the Volkswagen Arena as Anthony Martial handed them a 10th-minute lead but the Red Devils' poor defending at set-pieces cost them dearly with Naldo equalising shortly after.
Julian Draxler set up Vieirinha to round off a sublime team move and put Wolfsburg 2-1 ahead just before the half hour. But United clawed their way back into the game eight minutes from time when Josuha Guilavogui headed into his own net.
"We're disappointed. But if you look back we should have won our last game at home to PSV," lamented midfielder Marouane Fellaini, talking to BT Sport.
Winger Jesse Lingard, who saw a first-half effort ruled out for offside, added: "Over the whole course of the competition, we haven't done well enough to qualify."
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There were unsettling scenes after the final whistle in Wolfsburg, though, as fans were forced to stay inside the ground until the all-clear was given after a suspect package was found in the stadium's car park.
But the Dutch side hit back just two minutes later through Luuk de Jong and completed the turnaround with five minutes to play as Davy Propper earned the home side a 2-1 victory, ensuring they went through as runners-up behind Wolfsburg.
Manchester City wrested away top spot from Juventus in Group D as Manuel Pellegrini's side claimed a thrilling 4-2 victory over Borussia Moenchengladbach, while last year's finalists succumbed to a 1-0 defeat at Sevilla.
However, two quick-fire goals from England international Raheem Sterling put City back on track with Wilfried Bony sealing the win.
In Seville, Fernando Llorente's 65th-minute goal guided the Spaniards past Juve and enabled them to pip Gladbach into third, ensuring Unai Emery's men will have a chance to land a third consecutive Europa League crown.
Cristiano Ronaldo smashed his own pool stage scoring record as he netted four times in a crushing 8-0 victory over Malmo in Group A.
Troubled French striker Karim Benzema also scored his first hat-trick for five years, while Mateo Kovacic added his first goal for the Spanish giants.
"Five victories in a row, we are on a good run and need to continue like this," said Ronaldo.
"On a personal level, I am happy for another record."
Paris Saint-Germain eased to a 2-0 victory at home to Shakhtar Donetsk with second-half goals from Lucas and Zlatan Ibrahimovic at the Parc des Princes.
Atletico Madrid wrapped up first place in Group C as they avenged an earlier home loss to Benfica with a 2-1 victory in Lisbon.
Saul Niguez's first-half strike put Diego Simeone's men in front before Luciano Vietto netted a second on 55 minutes, with Kostas Mitroglou grabbing a consolation for the hosts.
Galatasaray denied Astana a famous first group stage win and the chance to continue their continental adventure in the Europa League as Selcuk Inan cancelled out Patrick Twumasi's opener in a 1-1 draw in Turkey.