Neal Maupay earned Albion a deserved point at the London Stadium with a stunning 90th-minute goal.
The Frenchman scored his fifth goal of the season with a brilliant overhead kick in the 89th minute to preserve Albion's unbeaten record in nine Premier League meetings with the Hammers.
Tomas Soucek headed in after just four minutes and West Ham also hit the bar and had a goal disallowed for offside after a VAR intervention at the London Stadium.
But Albion should have been level before Maupay’s dramatic intervention. He missed an early chance and Jakub Moder and Adam Lallana also went close in what was a thoroughly entertaining tussle. Albion have now drawn eight of their last ten games.
Graham Potter had made two changes to the side which drew 0-0 with Leeds on Saturday. Tariq Lamptey and Pascal Gross dropped to the bench with Jeremy Sarmiento coming in for his first Premier League start and Lallana recalled to midfield.
In the pre-match build-up Potter had warned about West Ham’s set-piece threat and it took them just four minutes to demonstrate it. Soucek found half a yard of space in the six-yard box to meet an in-swinging corner from Pablo Fornals and his flicked header flew past Sanchez.
Albion had actually made a decent start and Maupay dragged a shot from eight yards wide after they had pressed high and won the ball back from Said Benrahma on the edge of the West Ham box.
But Sanchez had to save a stinging drive at the second attempt when Vladimir Coufal let fly from distance and Albion suffered another setback when Sarmiento limped off after 12 minutes and was replaced by Solly March. Sanchez then picked out Bowen instead of a team-mate with an attempted clearance and was grateful to see the former Hull City striker's ambitious effort from a yard inside Albion territory sail wide as the keeper scrambled back towards his own goal.
West Ham went even closer in the 28th minute when Fornals struck the underside of the bar with Sanchez beaten after Michail Antonio had nodded Bowen’s cross into his path. The visitors were then forced into a second change when Adam Webster limped off and Shane Duffy replaced him.
Albion’s best chance of the first half came in the 39th minute when they should have levelled. A great move started by Lallana saw March and Maupay combine before Maupay’s perfect pass found Moder eight yards out, but Lukasz Fabianski saved his Polish compatriot’s drive with his feet and Lewis Dunk powered his header from the resultant corner a yard wide.
Dunk threatened again from a corner two minutes into the second half and this time his powerful header from Moder’s delivery was on target – but straight at Fabianski.
Moments later West Ham thought they had doubled their lead when Antonio bundled the ball over from a yard out after Albion didn’t clear another Fornals’ corner. But after a lengthy VAR check Chris Kavanagh ruled the goal out for offside, although the suggestion was that Albion had been appealing for a foul by Craig Dawson on Lallana in the build-up.
It was threatening to turn into a frustrating night for the 3000 Albion fans behind the goal their side attacked in the second half. On 68 minutes Albion prised open West Ham again with Trossard’s pass finding Lallana 12 yards out. He had time to take a touch but elected to strike it first time and his effort went a couple of yards wide.
Potter brought on Lamptey for the last 20 minutes with the game increasingly being played in the West Ham half. Albion continued to put together some pleasing passing moves but on several occasions the ball into the box didn’t find a blue and white shirt.
West Ham were a still a threat though on the break and from set-pieces and Sanchez made a superb save diving to his right to beat out Declan Rice’s powerful drive from the edge of the box after Duffy’s pass out was intercepted by the Hammers’ captain. The Spaniard was at it again a couple of minutes later with a flying save to his left after Antonio had met a deep cross with a powerful header.
The value of those saves became evident in the last minute of normal time. Lallana had limped off leaving Albion with ten men for the closing stages but they kept probing. Lamptey made a yard of space to cross from the right and Maupay instinctively sent an overhead kick under pressure from two defenders flying past Fabianski.
There was still time for Sanchez to make another save to deny Benrahma and Albion had to defend a couple of corners before they could celebrate a hard-earned point with those ecstatic Albion fans.
Albion: Sanchez, Veltman (sub: Lamptey 69), Dunk, Webster (Duffy 36), Cucurella, Bissouma, Moder, Sarmiento (March 12), Lallana, Trossard, Maupay.
Subs not used: Steele, Mac Allister, Mwepu, Gross, Locadia, Burn.