Iliman Ndiaye struck Everton's opener before Grant Xhaka found an equaliser for Sunderland in the opening minutes of the second-half.
Moyes told evertontv: "Yes, we want to keep it going. That's why I'm disappointed tonight, because I thought the second-half performance was way below the levels we've set generally, and I think we'd set the levels we think we're at in the first half for long periods. But it's not good enough if we're only doing it for 30 minutes.
"We need to be able to play well. We played quite well at Man City for 30 minutes, we played quite well against Tottenham for 30, 40 minutes, but it wasn't enough and it's not enough. We've got to find a way out – doing it for much longer periods than we're doing at the moment."
The Blues boss added: "After minutes or so I was thinking, 'We've got this game, we're playing so well and if we keep this up then we'll be getting the goals'. We get one, hit the post twice, then Barry misses a big chance which would have made it 2-0. I thought that took the wind out of our sails a little bit.
"I think everybody sort of went, 'Oh my goodness, we didn't take the chance we got', because Sunderland are one of the best defensive teams in the league at the moment and not giving away too many opportunities.
"So to have got 2-0 up would have been a big thing for us and we didn't get it. I thought they became the better team in the last 15 minutes of the first half and they were certainly the better team for the majority of the second half. I thought we were absymal in the second half. We were resilient enough to stick and get a point out of it, but I'm a bit disappointed. I'm more disappointed with the performance today than I am with the point."