Manager David Moyes was staring down the barrel of a defeat to his former club, but like he did against Brentford towards the end of February, O'Brien was on hand to earn a share of the spoils for the Toffees.

Tomas Soucek, who netted in West Ham's 3-1 win in the corresponding fixture last season, broke the deadlock with a fine finish in the 67th minute at Goodison Park.

The Toffees laid siege to the West Ham net after going behind, and almost levelled through Jack Harrison in the 74th minute, but he saw his effort blocked by the visiting defence as it aimed to see out just a fourth win under Graham Potter.

But Everton's pressure told when O'Brien headed home from Idrissa Gueye's volleyed centre in the 91st minute, and the hosts could then have won it with virtually the last kick of the game, only for Carlos Alcaraz to drag wide.

The draw moves Everton above Manchester United, which plays on Monday (AEDT), into 14th, two places above West Ham courtesy of a better goal difference.

"Brilliant character," Moyes said after the game. "[The point was] deserved in the end. We did quite well in the first half, but didn't create enough chances.

"Second half, West Ham started better. We made a few changes, the boys came on and made a difference. We got our reward. Maybe, a draw on the day was the right result.

"We know we lack quality in areas and are having to improve. If you can't take three [points], you have to make sure you get one. It's good we're beginning to get back to better levels. We know we have to pick up the levels and quality."

Moyes also paid tribute to O'Brien.

"He's done really well. He's a young centre-back developing at right-back. He's got some really good attributes," the Everton boss added.

"What he needs is Premier League experience. He came up against top players today in Bowen and Kudus and did well."

O'Brien thanked the home support for its contribution on what was a critical result for the Toffees.

"We were flat for 20 minutes and that's when they caught us, but then we got together and chased the game," O'Brien said.

"The crowd were amazing. The fans pushed us until the end. The last couple of games, we got draws we could have turned into wins. But we need more points. We want to climb the table more."

West Ham boss Potter could only focus on the positives after the draw.

"It's football, sometimes. When the environment turns here and the game changes as a result, it's difficult," he said. "Overall, a strong performance, it is what it is.

"We defended really well. I can't remember any chances they had. It was more pressure from their fans and balls into the box.

"We've had to be that way because of where we were to start with. It's a good point against a team in a good place. Every game is really important for us. We need to focus and go for the three points in every game."

"It's always very difficult for the team and fans when we're winning and concede in stoppage time," Soucek added. "We have to be disappointed."