Burnley boss Sean Dyche said: "It's rare in the Premier League to go away from home and have as much possession as we did.
"There's a lot of things right about the team at the moment but we've got to find the margins. We're close but we've got to make sure we do it rather than waiting for it to happen.
"After eight games we had two points. So we've been effective. But we need to get more effective."
"We're not the old West Ham any more, we're the new West Ham," said the Scot. "And we're going to try to become even better.
"It needs time and there will be periods when it's not going so well. There'll be periods when it is, but we're certainly not settling for where we are."
The Hammers have now kept four successive clean sheets in all competitions, and Moyes added: "We are pleased with that because I don't think you win games without them.
"You always have a chance of picking up a point if you don't concede a goal, and that was very good."