The Red Devils dominated the Cottagers at Old Trafford, controlling possession and delivering a record number of crosses into the box.
They were, however, to fall behind early on to a Steve Sidwell strike and took until 12 minutes from time to find a route back into the game.
Goals from Robin van Persie and Michael Carrick swung the match back in their favour, but there was to be a late twist in the tale as Darren Bent grabbed a stoppage-time leveller.
Moyes admits United should not have left themselves in such a position, with the struggles which have plagued them on home soil this season returning to haunt them once again.
He told Sky Sports: "We should have won it but lapses cost us. We probably had to defend three or four times in the whole game, but that happens sometimes in football.
"We tried everything to try and win - we crossed it, passed it, tried to get through - but we just didn't quite have the finishing touches.
"I think if we had got a goal early on then it would have completely altered the game. As it was, the more we went to score, they had a couple of chances and one of them they scored from. But I hardly felt as if they were on the park today.
"It has been quite often here at Old Trafford when we haven't got the goal, we have then maybe lost a goal and ended up chasing the game. We did it today.
"If we hadn't got back into the game then there would have been something up because we totally deserved it. We deserved to win the game, but you don't always get what you deserve."
United are now nine points adrift of the top four and in danger of missing out on a place in the UEFA Champions League.
Asked if the Red Devils can still break into that elite group, Moyes said: "We'll do our best."