Nikica Jelavic hit the woodwork twice in a decent all-round display from the hosts, who had gone ahead early through Steven Pienaar before succumbing to a Frank Lampard brace.
"We're not disappointed with the way we've played because we played really well and we've run the European champions really close. But we didn't quite have enough in the end," said Moyes.
"We had played well enough to get a second goal but had been unfortunate, having shots back off the woodwork. I thought we were a bit unlucky.
"We tried to make what we had work the best we could today and for long periods I thought we did, we played well enough to get something from the game.
"We were limited today with what we had, with Phil Neville and Darron Gibson out injured and Marouane Fellaini suspended.
"But Nikica had a great chance to make it 2-2 in what felt like the 94th minute."
Moyes apologised for confronting referee Howard Webb at the final whistle, adding: "I thought it was a free-kick on Leon Osman on the edge of the box in the 92nd minute by Lampard and he didn't give it.
"But I've apologised to Howard Webb because I shouldn't do that coming off the pitch."