Moyes was frustrated his side had not taken at least a point from the game having gone ahead so early and then seen Nikica Jelavic hit the woodwork twice before missing an added-time chance to snatch a draw.
"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, not having Phil Neville and Darron Gibson (through injury) and not got Marouane (Fellaini) because of suspension.
"But Nikica had a great chance to make it 2-2 in what felt like the 94th minute."