It looked like the Blades had failed to take advantage of a much-improved second half when Jean-Ricner Bellegarde's 89th-minute strike deflected past Wes Foderingham to cancel out Cameron Archer's brilliant opening goal.
But home players who were angry about how Wolverhampton Wanderers' equaliser came about pushed again for a winner and when Jones awarded a penalty for Fabio Silva's foul on George Baldock, Oliver Norwood kept his cool through a long video assistant referee review before smashing in the penalty for a 2-1 win.
Afterwards Gary O'Neil was furious with the decision against striker Silva, but Heckingbottom said he was unhappy with the way Wolves' goal came about.
"I don't think you'd have seen me if we hadn't been given that penalty, I might have been in prison somewhere," joked Heckingbottom, who was booked for his protests.
"It was the part Rob played in it, not allowing us to get any treatment on for Gus (Hamer). For (Rayan) Ait-Nouri he stopped the game.
"The only time we were allowed treatment on was when Tommy (Doyle) kicked the ball out for us, for Rhian (Brewster).
"Gus is injured, the ball's out of play and he said if he wants treatment he has to go off. We wanted to make the change, we wanted him to get treatment.
"He goes down again and he doesn't allow treatment on again and they get the equaliser. It's disgraceful, really."
The victory was Sheffield United's first of the season, quadrupling their Premier League points tally in a stroke. The way it came about added to the considerable psychological value for an under-pressure team and manager.
The Blades defended very well in the first half but only added an attacking threat in the second, which they dominated.
"The dramatic nature of it makes it feel even sweeter but it would have been harsh if we'd not (won), given how we performed, especially in the second half," argued Heckingbottom.
"I thought the first half was pretty tepid, both teams feeling each other out. Every time we won the ball back in dangerous areas, five yards over the halfway line, in the first half we generally gave it away within two passes. In the second half we didn't, we had much more quality and it made Wolves defend. That was the difference.
"What it shows and I can reiterate the message is you need cool heads. It's a tough, tough league and there's going to be games when we're not in them, the opposition take it away from us.
"I've certainly not panicked in those games but I know it doesn't feel good."
O'Neil was adamant the penalty for Silva's tackle from behind on Baldock should not have been a spot kick, but honest enough to admit it was more his team's own fault than Jones' that they failed to win.
"It's not a penalty," he said. "(There's) hardly any contact. It's a tough one to discuss, really.
"It's another terrible decision, I don't understand why he feels the need to give it – just pause, take your time, it's not about you, and if you make a clear and obvious error VAR will tell you after.
"I've been with him and he still thinks it's a penalty. He says there's contact – the only contact I can see is maybe Fabio's shinpad with Baldock's calf, but a graze. Baldock's already on his way down.
"I'm sat there watching it with him and he still tells me it's a penalty which tells me we're in a pretty bad spot because that's never, ever a penalty. If he gives a penalty for that or gives a foul for that much contact there'd have been one million fouls in the game.
"Because of the wording we use VAR can't intervene because it's not deemed clear and obvious because Fabio's shinpad has touched Baldock's calf. Craziness."
But he did add: "The first half we should be at least two in front, probably. I thought we had complete control and were very threatening and it's really disappointing we didn’t take those chances.
"We need to be better so refereeing decisions and things like that don't affect us. We should be 2-0 in front defending that situation.
"Second half I was very disappointed, Sheffield United picked up the injury just before and changed them from a front two to a front three.
"We spoke about it but the lads didn't recognise it really and we made too many bad decisions with the ball and created some negative momentum so their crowd were able to get behind them.
"First half we had three or four real good looks at their goalkeeper and decided not to shoot or miscontrolled at the last moment so we needed to be better in front of goal.
"Second half we needed a better understanding and we need the officials to be better."
Heckingbottom said the problem which led to Rhian Brewster being substituted just before half-time was a jarred knee, rather than a recurrence of the hamstring problems which have been such an unfortunate feature of his Bramall Lane career.
"No one seems too concerned," he said, when asked about the extent.