Goals in either half from Billy Sharp sealed United’s fourth successive victory, and seventh in nine league games under their manager Heckingbottom.
Albion finished with 10 men after captain Jake Livermore was sent off in the first half, a decision that Baggies boss Steve Bruce labelled “wrong”.
But Heckingbottom refused to get carried away, despite United having two games in hand on sixth-placed Nottingham Forest and a host of fellow promotion contenders still to come to Bramall Lane.
“Serious is Fulham, they’re running away with it,” Heckingbottom said.
“The rest of the league is crazy. We’re happy with where we are. We can’t get past 10th, we keep winning and can’t get past 10th.
“But we’re a point behind the play-offs with our games in hand, but that doesn’t necessarily mean anything. It means our run in is possibly harder than the others, but that’s where our squad comes in.
“Six of the top teams have to come and play here and they’ll see it as a chance to come and take points off us. We can only look after ourselves.”
Bruce thought that his side were on top until Livermore saw red, for scything down Conor Hourihane.
“I was pleased with the way we started, but I thought the red card was harsh. It was a decent challenge,” he said.
“There was a tackle in front of me not long earlier and the referee didn’t even give a free-kick.”