Three goals in seven minutes looked to have secured the most unlikely of comebacks for Manchester United only for Sheffield United to rescue a point at the death.
Fleck fortuitously bundled home the rebound from a David De Gea save to give the hosts a well-earned lead on 19 minutes before Mousset struck home seven minutes after the break.
Teenager Brandon Williams' stunning strike gave United hope on 72 minutes and substitute Mason Greenwood tapped home within minutes of coming on to equalise.
Just two minutes later, Marcus Rashford looked to have sealed an incredible comeback when he slotted home with 11 minutes left.
But a last-gasp equaliser from substitute Oliver McBurnie survived a VAR check to clinch a remarkable equaliser to deny United back-to-back away victories in the league for first time since February.
The hosts were forced to play reserve goalkeeper Simon Moore in goal with on-loan Dean Henderson ineligible to face his parent club, while Phil Jones came in for his first Premier League start of the season in a back three.
But he had very little to do as the home side came bursting out of the blocks. Indeed, it was his opposite number De Gea who was forced into action with an outstanding double save to deny John Lundstram's volley and then a close-range header from David McGoldrick.
But Chris Wilder's side found the breakthrough shortly after. Mousset shrugged off the challenge of Jones and cut it back to Lundstram, who saw his effort saved by Fleck bundled home.
United's lifeless display meant they were fortunate to go into the break only a goal down but it got worse when Mousset, played in by Fleck this time, fired low into the corner from 20 yards to double their advantage.
The visitors looked down and out until Williams' fine strike on the volley gave them hope with 18 minutes to go. Eighteen-year-old Greenwood came off the bench and within three minutes, was on the scoresheet, tapping home from Rashford's cross to make it 2-2.
Rashford made it 3-2 just 90 seconds later to cap a remarkable turnaround, slotting home from Daniel James' cross but there was still time for more drama when McBurnie volleyed home in the 90th-minute. After a lengthy VAR check, the goal was allowed to stand and Man Utd were denied victory - a scenario that looked almost impossible with 20 minutes to go.
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