The Scot believes his side did enough to beat Mauricio Pochettino's high-flying Saints but paid the price for a lack of potency in front of goal and some poor defending
The Red Devils spurned a number of chances while 1-0 up against Mauricio Pochettino's side before Adam Lallana's 89th-minute equaliser meant the champions have now failed to win at Old Trafford on three out of four occasions in the league this term.
Robin van Persie gave Moyes' side the lead midway through the first half, but the former Everton manager was frustrated his side could not put the game to bed.
"We were disappointed to lose a goal so late on. We had good opportunities to get the game finished off, but while it was 1-0 there was always a chance Southampton could get something," he told reporters.
"I thought we’d just about seen it out when they got the corner late on.
“We didn’t do well enough [with the goal] but we should have put the game to bed earlier on. We had a couple of chances and hit the bar.
"We had some pockets of play that were very good and others where we weren’t so good. It was a mixed game but we still had those chances which I’d hope we could finish off.
“We’ve had opportunities to get the goals and see it off, but we’ve got to do better. But we’ve got to do better at the other end too and make sure we’re harder to score against."
Van Persie's strike was the first United have scored at home in the league from open play this season, and with Chelsea and West Brom also coming away from Old Trafford with points, there has been talk that sides now come to the ground without any fear.
But Moyes insisted that this had nothing to do with the retirement of Sir Alex Fergsuon, believing it is down to the players to make the stadium a fortress again.
"The fear comes from the team on the pitch," he added. "The players are the thing because of the quality. Obviously Sir Alex has great history, but the players are always the ones who have to do it on the pitch."
Meanwhile, Pochettino was thrilled with the result and believes that this draw, coupled with already winning at Anfield, will give his young side great belief heading into the rest of the season.
“Getting a very good win at Anfield and a positive result today is amazing for this team," he said.
"It definitely certifies the quality of this project and how we move forward, especially for the younger players who will gain so much confidence from this.
“We want to keep on believing that we can do well. It’s a very positive run we’re on at the moment and the sky’s the limit."