Just minutes into the game Jermain Defoe unleashed a low 20-yarder that Clarets keeper Brian Jensen found too hot to handle, before Christian Dailly completely missed the subsequent corner six yards out.
The Hammers continued to press at any opportunity, and on 20 minutes their superiority finally prevailed. Having out-paced Arthur Gnohere, Defoe saw his low shot cannon off Jensen's out stretched foot before David Connolly pounced onto the rebound and coolly netted his eighth goal of the season from a tight angle.
Jensen was again looking exposed on the half-hour mark, but this time he stood large and parried Connolly's 18-yarder skywards. Up until then his counterpart James had been a mere bystander but on 38 minutes Ian Moore carved into the box, drew David James and presented Delroy Facey with a three-yard tap in.
As the interval approached Defoe wastefully blasted over and seconds after the restart Connolly rattled the bar before Jensen parried Matthew Etherington's stinging angled shot.
The Hammers inability to turn pressure into goals was frustrating the home fans as Ian Pearce sent his point-blank 64th minute shot thudding into Jensen's knees.
The Burnley keeper continued to form a one-man barricade and wholly against the run of play substitute Tony Grant popped up at the other end to wrap James right hand post in the 80th minute.
Seconds later, there was to be no similar let off for West Ham as Moore raced away and out muscled Thomas Repka to bundle the ball over the line after the advancing James had deflected the ball skywards.
But just when it looked like the Hammers had thrown it all away substitute Hutchison rescued that late point with an unstoppable 12-yard volley after Pearce nodded back across goal.