Albion had numerous chances in the first half but went behind when Jay Rodriguez netted from the spot against his former side.
Burnley had a second half shout for a spot kick turned down before hitting the post - but Albion dug in.
Jed Wallace spurned a superb chance and as the clock hit 98 minutes it looked as if the game had gone - but Thomas-Asante fired home from close range to hand Albion a point.
