Sir Alex Ferguson's side dominated the first 45 minutes but the Serie A leaders came back well after the break and had several good chances to take a lead to Old Trafford in two weeks' time.
It had been one of the most anticipated fixtures of the season to date with Ferguson coming up against the former FC Porto and Chelsea boss Jose Mourinho.
All the pre-match talk had been about United's injury crisis but Jonny Evans and John O'Shea were both fit enough to start while Wayne Rooney has to settle for a place on the substitute's bench.
United started very brightly and Inter's Brazilian goalkeeper Julio Cesar was forced to make an instinctive sixth-minute save to keep Cristiano Ronaldo's powerful header from Ryan Giggs' pinpoint corner out.
Two minutes later Ronaldo went close to breaking the deadlock again with a free-kick that whistled just wide.
It was all United and in the 19th minute Giggs sent Dimitar Berbatov free. The Bulgarian beat the offside trap but wasted a good opening with some lackadaisical play.
Eight minutes later Giggs did superbly to turn Maicon, who was badly out of position. The veteran Welshman headed towards goal but his drive was kept out by Julio Cesar's face.
Just before half-time Ronaldo stepped up to take yet another free-kick but again he was denied by Julio Cesar's swooping save.
The tide turned after the interval though with Inter creating all the openings. In the 48th minute Zlatan Ibrahimovic dummied a cross but Adriano miskicked when in a good position inside the penalty area.
Sixty seconds later Adriano took a tumble as he went for a cross. Rio Ferdinand had a hand on the Brazilian's shoulder but the referee refused to award a penalty.
In the 67th minute Ji-Sung Park was inches away from getting a toe to Ronaldo's cross but Inter almost won it with eight minutes left when Ibrahimovic flicked on a corner.
The ball hit Esteban Cambiasso, who was virtually on the goal-line, on the chest but it spun away from danger rather than ending up in the back of the net.
With the last kick of the game Ronaldo's free-kick hit Julio Cesar in the chest. It certainly couldn't be called an orthodox save but the Brazilian kept it out to ensure the match ended goalless.