Returning to Wearside for the first time since his spell in charge of Sunderland ended in relegation in 2017, Moyes watched his Everton side take the lead through Iliman Ndiaye’s superb solo goal before losing control of the match after half-time.
"We were abysmal in the second half," Moyes said. "In the opening 25 minutes we should have been a couple of goals up but we couldn't get that second goal." He added: "For 20-30 minutes I thought I'd be disappointed going home with a point. By the end of the game, I was pleased we got a point from it."
The Toffees dominated the first half, with James Garner firing wide just nine seconds in and Jack Grealish striking the post before Ndiaye’s dazzling opener in the 15th minute.
However, Thierno Barry’s missed chance to make it 2-0 in the 28th minute, firing over from close range after a Grealish cross, proved costly as Sunderland came out revitalised after the break.
Sunderland captain Granit Xhaka inspired the comeback, levelling the match just 42 seconds into the second half with a deflected strike from the edge of the box that looped off James Tarkowski and left Jordan Pickford wrong-footed. From that point, the home side looked the stronger team, forcing Everton to defend deep.
"We missed a big chance to make it 2-0 and because we don't get that, we sat back and it gave Sunderland a bit of confidence, gave the crowd something to get behind," Moyes reflected on Barry's miss.
"We didn't manage the last 15 minutes of the first half particularly well. We certainly didn't manage long periods of the second half that well."
Despite the dropped points, Moyes praised Ndiaye, who was substituted in the 62nd minute after limping off with what appeared to be a tightening injury in his leg.
Everton boss confirmed post-match that he does not believe it is serious and was a precautionary move. Ndiaye's goal was a highlight, as he evaded four defenders before curling into the corner, a moment of brilliance that showcased his standout form this season.
Everton, who sit 14th in the table with 11 points after the draw, will look to rediscover their winning touch when they host Fulham at Goodison Park on Saturday.
Manchester City secured a second successive Premier League victory at home against Sunderland, Chelsea were held scoreless away to Bournemouth while Tottenham managed to beat Brentford.
Aston Villa delivered a major jolt to the Premier League title race as Emiliano Buendía struck deep into stoppage time to seal a 2-1 win over Arsenal on Saturday.
David Moyes urged Everton to stay on the front foot as they prepare for Saturday's Premier League meeting with Nottingham Forest at Hill Dickinson Stadium, where Sean Dyche made his first return since leaving the club nearly a year ago.
Mikel Arteta has said that Aston Villa's blistering run of form has made them genuine contenders in the Premier League title race, as Arsenal prepare for a high-stakes trip to Villa Park.
