The Blades were thoroughly deserving of a 2-0 advantage midway through the second half before the Red Devils hit three goals in a blistering seven-minute spell to lead.
Just as it appeared Man United were going to leave with maximum points, McBurnie volleyed in from six-yards and, following a VAR review to clarify if he handled or not, the goal was allowed to stand.
It was just reward for Chris Wilder’s tenacious side, who bested their visitors for long spells of the contest.
John Lundstram drew a fine save from David de Gea early on, while David McGoldrick then saw a close ranger header palmed out by the United stopper.
Sheffield United didn’t have to wait long for their rewards however.
A long ball from the back gave Lys Mousset a chance to hassle Phil Jones and the defender lost the physical battle.
Lundstram’s shot from the resulting cross was saved by de Gea, but John Fleck was there waiting and the ball rather cannoned into the net off him.
Manchester United offered very little going forward in the first half and Ole Gunnar Solskjaer hooked Jones at the interval, reverting from his three-man defence as he introduced Jesse Lingard.
Within seven minutes of the restart United were in big trouble. Fleck fed Mousset and he strode purposefully towards goal before curling a low shot into the corner from 25-yards.
The hosts were in cruise control by the hour, with United’s front three of Marcus Rashford, Anthony Martial and Daniel James seemingly isolated and lacking ideas.
It took youngster Brandon Williams, up from full-back, to ignite a fightback, ghosting in at the back post and sliding a terrific low shot on the volley beyond Simon Moore in the Blades goal.
That gave belief to the previously lacklustre visitors and they were duly level five minutes later when sub Mason Greenwood got the finishing touch with a sliding finish from Rashford’s cross.
Buoyed, Solskjaer’s side completed a dramatic turnaround two minutes later. James made inroads on the left and his cutback was perfect for Rashford, who applied the finish with Moore stranded.
To their credit, the Blades came back fighting in the final ten minutes, while Man United took Martial off in favour of Axel Tuanzebe in a bid to shore things up.
They were closing in on the win, but with 10 seconds remaining in normal time McBurnie controlled a pass from Callum Robinson on the six-yard line and volley into the corner, de Gea’s outstretched arm merely able to divert the ball into the net.
There was a hint of handball as McBurnie controlled, but replays were inconclusive and – as he was clearly attempting to get his arm away from the ball – the VAR verdict did not chalk the goal off.
A six-goal Sunday thriller that will leave both camps feeling frustration but, in the case of the home side at least, a fruitless end really would have been a travesty.