Aguero fires Man City to win as Southampton climb table
Pep Guardiola's team, with defender Aymeric Laporte back in the side, looked as though they might rue Gabriel Jesus's squandered first-half penalty but their Argentine hitman came to the rescue.
Despite the win, City remain 13 points behind runaway leaders Liverpool, having played two games more and their bid to defend their title appears hopeless.
Another victory for Southampton lifted Ralph Hassenhuttl's side to ninth in the table while Nigel Pearson saw his in-form Watford fall to a late Aston Villa goal in a 2-1 defeat.
Elsewhere, Moise Kean scored his first goal for Everton in their 2-2 draw against Newcastle while Bournemouth beat Brighton to give themselves fresh hope of survival.
Manchester City's French central defender Laporte has been out of action since suffering a knee injury at the end of August and he was a surprise inclusion at Bramall Lane.
Sheffield United goalkeeper Dean Henderson denied Jesus from the penalty spot about 10 minutes from half-time after John Egan brought down Riyad Mahrez.
The Brazilian has now failed to score three of his five Premier League penalties.
But City made amends when substitute Sergio Aguero, just on for Jesus, tapped in from Kevin De Bruyne's cross in the 73rd minute.
Southampton's 2-0 victory against Crystal Palace continued a remarkable turnaround for the club, who hit rock bottom earlier this season in a 9-0 defeat by Leicester.
Nathan Redmond controlled Jack Stephens' ball forward, turned away from Martin Kelly and rifled into the top corner from 20 yards midway through the first half.
Southampton went 2-0 up three minutes into the second half thanks to a fine Stuart Armstrong goal from distance.
Watford took the lead against Villa when Troy Deeney headed in Gerard Deulofeu's cross seven minutes before the break but the home side hit back in the second half through Douglas Luiz.
Tyrone Mings scored a dramatic late winner for Villa deep into stoppage time.
Kean scored his first goal for Everton since his arrival from Juventus in the 30th minute of their match against Newcastle.
Dominic Calvert-Lewin doubled the lead in the second half but Newcastle staged a remarkable comeback, with Florian Lejeune grabbing two stoppage-time goals.
Bournemouth, who had lost their past four matches, all against relegation rivals, beat Brighton 3-1 thanks to goals from Harry Wilson, an own goal from Pascal Gross and a second-half Callum Wilson effort.
Jurgen Klopp's Liverpool are in action on Thursday at Wolves while Manchester United take on Burnley without Marcus Rashford, who could be out for three months with a back injury.