In the 75th minute, the hosts seized the lead as substitute Sinclair Armstrong sprinted onto Max Bird's perfectly placed through-ball. However, just as Armstrong was about to round goalkeeper Michael Cooper, he was tripped, earning the team a crucial penalty.
Another replacement, Anis Mehmeti, drilled the resulting spot-kick low to Cooper's right and, although the keeper guessed the direction correctly, he could not prevent the ball from nestling in the bottom corner.
The Blades' 86th-minute leveller came from a left-wing corner, unnecessarily conceded by City goalkeeper Max O'Leary's mistimed punch when he could have caught the ball. His defence failed to get a telling touch and yet another sub, Ryan One, rose to nod home.
The seven minutes of indicated stoppage time had been played when Burrows struck a sweet left-footed shot past O'Leary from inside the box to send the Sheffield United fans behind the goal into rapturous celebrations.
TrendingThe result ended City's eight-match unbeaten run and their misery was compounded by a red card for defender Rob Dickie for denying a goalscoring opportunity just before Burrows' winner.
The home side had to make a change after only six minutes, Mehmeti replacing the injured Mark Sykes.
With fireworks going off around the ground, the first half developed into something of a damp squib.
It burst into life briefly in the 25th minute when City's Nahki Wells shot inches wide of the near post from Yu Hirakawa's low cross from the left towards the six-yard box.
Seconds later, Gustavo Hamer broke clear at the other end and saw his low drive from an angle thump against the far post and back into play.
Those efforts apart, there was little to worry either goalkeeper in an opening 45 minutes full of attacking endeavour but littered with poor choices and final passes in the final third.
City made a change at the start of the second half, sending on striker Armstrong for Wells. They went close to taking a 56th-minute lead when Jason Knight's downward header from a Bird free-kick clipped a post.
Boss Chris Wilder responded with a triple change, introducing Ollie Arblaster, Kieffer Moore and Jesurun Rak-Sakyi for Sydie Peck, Tyrese Campbell and Andre Brooks. But it was City who were looking the more likely winners as Mehmeti's low drive from a narrow angle was saved with an outstretched leg by Cooper.
The goalkeeper had less difficulty dealing with a long-range effort from Marcus McGuane before Moore's brief contribution ended with injury and he was replaced by One.
City's second-half dominance was finally rewarded by Mehmeti's penalty. But after One's equaliser, Sheffield United finished the match strongly and created several openings before Burrows broke Bristol City hearts.
Bristol City's Liam Manning:
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Sheffield United's Chris Wilder:
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