The Hammers raced to a three-goal first-half lead after strikes from Jesse Lingard, Jarrod Bowen and Tomas Soucek before Alexandre Lacazette sparked a Gunners comeback.
Lacazette’s deflected strike off Soucek made it 3-1 before half-time, with Craig Dawson netting a 61st minute own goal before the Frenchman equalised with an 82nd minute header.
West Ham United would have moved level on points with Chelsea in fourth with a win, but instead remain fifth.
“We got a little bit sloppy after wed played ever so well for the first 30 minutes which resulted them getting a goal just before half-time which, when you see it, was going wide before it hit Tomas and goes in so that was hard to take,” Moyes said.
“Nevertheless, we had a good lead going in at half-time but the minute we lost that second goal we were feeling under pressure.”
The Hammers are coming under pressure from Tottenham now, with Spurs defeating Aston Villa 2-0 on Sunday to close within one point in sixth.
On the race for UEFA Champions League football, Moyes added: “We’re trying to be competitive right until the end of the season and we know we’re in a difficult run of fixtures.
“They’re all difficult, but we’ll have some fixtures near the end of the season which we might think we have a good chance in, and we had a really good chance today.
“You can’t deny Arsenal just beat Tottenham and they just got through to the quarter-finals of the Europa League, so they’re on a bit of a run at the moment and it was important to halt them so they couldn’t get any closer to us, so hopefully we can keep well in front of them.”
Spurs win over Villa was earned with goals from Carlos Vinicius, breaking his league duck, along with a Harry Kane penalty which he won himself.
The victory eases the pressure on Tottenham after their shock midweek Europa League exit to Dinamo Zagreb.
Spurs boss Jose Mourinho said: “This should not be a reaction, this should be a permanent state of mind, permanent soul in the team, I think that is our next challenge - to have this standard in terms of effort, commitment, sacrifice.”