A scrappy first half low in quality burst into life just before halftime when Issa Diop headed in a Robert Snodgrass free kick to put the hosts in front.
But Everton's response was immediate and Lucas Digne's corner was flicked on by Mason Holgate and Dominic Calvert-Lewin headed in at the far post for his 11th goal of the season.
West Ham manager Moyes, formerly the long-serving boss of Everton, has lifted the spirits since returning for a second stint in charge of the struggling Hammers.
That did not stop some fans staging a protest before kickoff about the running of the club, 10 years after it was bought by David Sullivan and David Gold.
Moyes, who saved West Ham from the drop in 2017-18 after replacing Slaven Bilic, knows he faces another scrap, especially when they struggle so often at home and fail to build on leads.
They have now dropped 17 points from winning positions this season, more than other side in the top flight, and remain only one point above the relegation zone while his old club are 11th.
"I think every point is important in this league. I thought we created one or two good chances but I thought it was even between the two sides," Moyes said.
"I thought the two or three minutes after we scored, we were slow to react and it looked like we were trying to play out to halftime and that left us very open."
While West Ham continue to flirt with danger, Everton's resurgence, begun by Duncan Ferguson as caretaker manager after Marco Silva's sacking, continues under Carlo Ancelotti.
"It's a good point," Italian Ancelotti said. "We did not play well first half, second half was better."
West Ham keeper Darren Randolph, who returned to the club this week from Middlesbrough, said that despite the loss of two points, there were plenty of positives.
"We were very solid and we limited their chances. We created a lot of chances in the first half. We are disappointed not to get the win," he said.