Anthony Martial put United ahead when he beat a high offside trap to corral a Juan Mata through ball and finish at the far post in the 10th minute.
But United were only ahead for two minutes when Naldo flicked on Ricardo Rodriguez’s free kick on the volley to beat David De Gea and level the score at 1-1.
Wolfsburg took the lead through a fine piece of play on 29 minutes, Julian Draxler playing a one-two to break into the United box before squaring for Vieirinha to tap into an open net.
United were given a lifeline when Josuha Guilavogui headed into his own net in the 82nd minute, but again the Premier League side were undone two minutes later when lacklustre defending allowed Naldo to score following a corner.
Wolfsburg won Group B while United settle for a spot in the Europa League.
PSV rallied to defeat CSKA Moscow 2-1 to finish runners-up and move on ahead of Man United.
In other results Cristiano Ronaldo scored four times and Karim Benzema added a hat trick as Group A winners Real Madrid routed Malmo 8-0.
PSG had already secured the runners-up spot in Group A and defeated Shakhtar Donetsk 2-0.
Raheem Sterling scored two and set up another goal as Manchester City produced a late surge to beat Borussia Monchengladbach 4-2 and top Group D.
Sevilla defeated Group D runners-up Juventus 1-0 to earn a spot in the Europa League.
Luciano Vietto had a goal and an assist to help Atletico Madrid win Group C with a 2-1 victory at Benfica, while Galatasaray drew 1-1 with FC Astana to secure their spot in the Europa League.