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.

Plymouth striker Oseni shocked by Nigeria call-up
African athletes possess NFL traits in abundance — Umenyiora
AFCON 2027: Udeze warns Chelle not to underestimate opponents
Joshua, Fury in contract stalemate
Agunbiade emerges Nigeria’s lone survivor at WTT Lagos
Nigeria call-up boosted my confidence – Alebiosu
First Eagles invite thrills Rangers’ Oputa
270 athletes begin journey to stardom in Edo
Itauma unfazed by Usyk threat
Arsenal win English Premier League after Man City draw at Bournemouth
AFCON qualifiers: Nigeria face Madagascar, Tanzania, G’Bissau
FIFA WORLD CUP 2026: Dreaming of Olise, Musiala, Eze, Saka - Priceless Jewels Nigeria Lost to Treachery
Rangers International going, going . . . (63,794 views)
Amaju Pinnick: A cat with nine lives (55,115 views)
Second Term: Amaju Pinnick, Other NFF Heavyweights Home to Roost •How Pinnick Broke the Jinx (53,071 views)
Current issues in Nigerian sports: Matters arising (52,636 views)
Sports Development: Zenith Bank on the zenith (52,550 views)
Missing $150,000 IAAF Grant: Solomon Dalung’s Hide and Seek game (52,455 views)
Gov. Abdullahi Ganduje’s solid footprints, commitment to sports development in Kano State (52,288 views)
NFF Presidency: Pinnick, Maigari, Ogunjobi, Okoye in Battle for Supremacy (51,849 views)
Olopade, BET9A wave of revolution in NNL (51,035 views)
Commonwealth Games 2018: Shame of Muhammadu Buhari, Solomon Dalung (49,532 views)
Ibrahimovic’s Man U exit: Whose decision is it? And in whose interest? (47,960 views)
John Mikel Obi: Segun Odegbami’s Outrageous Call! (47,406 views)