United went into Saturday's Premier League fixture hoping to avoid back-to-back home defeats for the first time since 2002 and, in the end, they can only blame themselves for not taking all the points.
Adam Lallana's goal in the last minute of normal time cancelled out Robin van Persie's earlier strike.
It was probably no less than the away side deserved but Moyes was still disappointed, especially after holding out for so long.
Speaking to Sky Sports after the game, Moyes said: "I am disappointed we didn't take the points. We had the chances.
"We should have wrapped it up. There were little pockets in the game where we had control. It's just a shame we didn't finish it off.
"But, there is always a chance when you are one goal up that you could concede.
"Overall though, I thought Jonny Evans and Phil Jones played really well at the back so I wouldn't blame them for the goal. We are looking for that consistency but we are struggling to find it at the moment."
The big news before the game was Adnan Januzaj's decision to sign a new five-year deal with the club and, understandably, Moyes was delighted.
He added: "It is really important for the club that he stays and we saw today with the goal what he is capable of. It was a wonderful pass."
The result means that Moyes's side have taken just five points from four home league matches this season as their Premier League title defence struggles to gather pace.
I've just checked the bench, sorry can't believe it all three Zaha, Kagawa and Hernandez on the bench and he didn't use one of them!!! Apologies for not doing my hoemwork, now I'm really depressed!- cantonagod