Burnley are facing a fight of their own to avoid the drop and things looked good when Graham Branch had them in front, but they were pegged back as David Connolly tucked home a penalty.
The two sides were closely matched throughout, with chances traded at each end.
Connolly should probably have scored when set free in the first half, but he was denied by a wonderful low save from Brian Jensen.
The resulting corner saw Scottish international Christian Dailly presented with a chance six yards out but he failed to make proper contact and Jensen saved comfortably.
Burnley have a cutting edge of their own and an incisive move sliced open West Ham's defence and Branch capped the fine build-up play with a superb finish.
The lead lasted just five minutes, however, as Connolly tucked home a penalty with ease after Bobby Zamora had been felled in the box.
West Ham hit the bar after the break, but it was Burnley who almost stole the points when Tony Grant saw a shot superbly saved by Stephen Bywater and Robbie Blake went within a whisker of winning the game when shooting low from the edge of the box.
