Tomas Soucek equalised from close range to cancel out Wout Weghorst’s opener as West Ham extended their unbeaten home run to eight games in all competitions.
The Hammers could have snatched victory late on but they found an inspired Nick Pope, who produced three big saves in the final stages, standing in their way.
Both sides started with attacking intent but the game took a turn for the worse when Westwood was stretchered off with a worrying ankle injury after what looked like a innocuous challenge with Nikola Vlasic.
Play was stopped for eight minutes with medical staff from both clubs attending to the Burnley midfielder.
Burnley interim boss Mike Jackson said: “He’s gone to hospital and we’re waiting for an update. We’re hoping he’s OK – our thoughts are with him.
“You could see it wasn’t good from the reaction of the players. He looked distraught, but fingers crossed.
“When you see Declan Rice come over and say it’s not good, you know it’s not. The lad himself put his hand up, and he doesn’t usually go down.
“It could be the end of his season, but there is a squad here ready to play. Josh Brownhill came in and has done really well.”
No-one was more visibly upset than Vlasic, who dropped to his knees with concern, but the incident appeared to galvanise the lowly visitors. Three minutes after Westwood’s departure, Burnley scored the opener.
Maxwel Cornet’s corner was met by Jay Rodriguez, who could only divert the ball against the crossbar, before Weghorst rose highest to head in the rebound.
Cornet had the chance to double the visitors’ lead after being brought down inside the area by Lukasz Fabianksi, but his poorly hit penalty went wide
The Clarets sacked boss Sean Dyche on Friday but remained resolute under the interim guidance of Under-23s coach Jackson and injured club captain Ben Mee until Soucek salvaged a draw for the hosts.
The result means Burnley move to within three points of safety, although they have played a game more than 17th-placed Everton, while West Ham remain seventh, five points outside the Champions League places.
Burnley made headlines on Friday after they dismissed Dyche, who was the Premier League’s longest-serving current manager, having taken over at Turf Moor in October 2012.
Two days later, the Clarets headed to an in-form West Ham side who had just booked their place in the Europa League semi-final.
The Hammers started brightly and Jarrod Bowen’s whipped effort was well kept out by an acrobatic save from the influential Pope.
