Stoke opened the scoring after a moment to forget for World Cup hopeful Hart in front of watching England manager Gareth Southgate as he failed to hold a shot from Xherdan Shaqiri and Crouch was in the right place to convert from close-range.
But West Ham roared back and salvaged a potentially invaluable point in the last minute of the 90 when Cresswell's ball into the box was met on the volley by Carroll, who guided the ball beyond a helpless Butland.More news: Russia's U.S. ambassador warns of consequences for Syria strikes
Stoke are five points from safety with only four games left to avoid slipping into the Championship.
Lambert reflected afterwards: 'It is a blow because we were so close to winning. We played good in parts of the game, it was just the fact we went one behind, because if anyone was going to score first I thought it would be us. "With the atmosphere in stadium, it is a hard place to come but I thought we were excellent". In the main, we played well and were on top when they got their goal.
"If we can do that (beat Burnley), it will set the cat right among the pigeons", said Lambert. The couple have been targeted since the Northumberland official, who was on duty at West Ham last night (above), gave Real Madrid a disputed penalty from which they knocked out Juventus in last week's Champions League quarter-final.
Meanwhile, the Hammers sit seven points clear of the bottom three, and have an extra game in hand.More news: West Brom 'pulling in same direction' for Darren Moore - James McClean
Marko Arnautovic had already had a header ruled out for offside when Edimilson Fernandes rifled into the net on 67 minuets.
Police have warned social media trolls they could be committing a criminal offence by abusing referee Michael Oliver and his wife Lucy.
And after West Ham's equaliser, Javier Hernandez had a goal disallowed for a foul by Carroll.
"We had a few chances, goals disallowed, but for us, it's a good result".More news: Politicians preoccupied with 2019 poll but I'm bothered about security