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.

Everyone expects Spain to beat us, says Belgium coach
Okoye backs France to win World Cup
Joshua, Fury opponents vow to ruin mega-fight plans
C’wealth Games: Olopade targets swimming, judo breakthrough
Teen girl dies celebrating France’s World Cup win
Balogun apologises to fans after US W’Cup exit
Eaglets suffer second defeat in China tourney
Alalshikh sets condition for Joshua, Fury Wembley fight
France urges calm during World Cup game against MoroccoFrance urges calm during World Cup game against Morocco
2026 FIFA WORLD CUP FINALS: Unprecedented Controversies Simmer as Stakeholders Brand FIFA 'An Absolute Disgrace'
Boniface salutes Neymar, Ronaldo after World Cup exits
Durodola gets 45th KO win at National Stadium
Rangers International going, going . . . (63,945 views)
Amaju Pinnick: A cat with nine lives (55,283 views)
Second Term: Amaju Pinnick, Other NFF Heavyweights Home to Roost •How Pinnick Broke the Jinx (53,191 views)
Current issues in Nigerian sports: Matters arising (52,801 views)
Sports Development: Zenith Bank on the zenith (52,686 views)
Missing $150,000 IAAF Grant: Solomon Dalung’s Hide and Seek game (52,553 views)
Gov. Abdullahi Ganduje’s solid footprints, commitment to sports development in Kano State (52,442 views)
NFF Presidency: Pinnick, Maigari, Ogunjobi, Okoye in Battle for Supremacy (51,961 views)
Olopade, BET9A wave of revolution in NNL (51,177 views)
Commonwealth Games 2018: Shame of Muhammadu Buhari, Solomon Dalung (49,645 views)
Ibrahimovic’s Man U exit: Whose decision is it? And in whose interest? (48,123 views)
John Mikel Obi: Segun Odegbami’s Outrageous Call! (47,512 views)