The Japan international settled an enthralling tie with a superb 81st minute free-kick and blew Group F wide open in the process.
The Bhoys secured their passage into the last 16 for the first time in their history and United must now get something from their trip to Benfica next month if they are to join Gordon Strachan's side in the knockout stages.
Louis Saha could have grabbed a vital point for United in the dying stages, but the Frenchman saw his 90th penalty saved brilliantly by Artur Boruc after Shaun Maloney was harshly adjudged to have charged down Cristiano Ronaldo's free-kick with his hand.
But the night belonged to Celtic who - despite being on the back foot for most of the match - soaked up wave after wave of United attacks and then hit the Premiership leaders with a spectacular sucker punch.
Nemanja Vidic was adjudged to have fouled substitute Jiri Jarosik 30 yards from goal in a decision which angered United.
And Nakamura rubbed salt into the wounds as he lined up the set-piece and executed it perfectly with his left foot over the top of the defensive wall and beyond the despairing dive of Edwin Van der Sar.
Earlier - in a match that the visitors dominated in terms of possession - Wayne Rooney dragged an 18-yard volley from Ryan Giggs' corner wide of the mark on the half-hour, before Boruc charged off his line to block a low shot from Ronaldo.
At the other end, Gravesen might have headed Celtic into a 45th minute lead as the home side broke quickly from defence and fired a warning to a then dominant United.
After the break the visitors continued to hold the upper hand, but for all their attacking play lacked the quality in the final third to open up an increasingly confident home defence.
And after Saha and Rooney had both spurned half-chances, Nakamura stepped up to produce the only real piece of attacking quality on a night to remember in Glasgow.