Iliman Ndiaye scored the winner to earn a big three points for the Blades, on an afternoon when play-off rivals Luton Town, Middlesbrough and Millwall all drew.
That leaves United three points ahead of seventh-placed Millwall with just two games of the regular season remaining, and surely in pole position to qualify for the end-of-season promotion shootout.
“I’ve not seen other results but I’m just bothered about us – and us winning,” Heckingbottom insisted after the game.
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“While there is only one goal in it, it’s always going to be nervy but I thought we stood strong. Wes mopped things up which came in the box without us really suffering any shots on target which is great.
“We could have made it easier for ourselves when we broke away a few times in the second half. We could have put the game to bed but it’s not going to be like that this season. We keep getting things thrown at us or things we have battling against, so you have to battle on.”
Heckingbottom won promotion via the play-offs as a player with both Barnsley and United’s city rivals Wednesday, and again as a manager with his boyhood club.
"Our first goal is to try and get in the play-offs,” Heckingbottom added.
“They're brilliant occasions and we're chasing that, we want to feel that.”