Haaland hit the post with a penalty attempt in the first half before his header put City ahead after the break at Bramall Lane.
City were on the verge of dropping points when Jayden Bogle levelled against the run of play with just five minutes remaining.
Rodri saved City with a stunning strike three minutes later, sending Pep Guardiola’s team to the top of the table after three victories in a row.
City’s latest triumph came without Guardiola, who was watching on television while recovering from back surgery in Barcelona.
The City manager was in close contact with his deputy Juanma Lillo, who took over at Bramall Lane and will also be in charge for Saturday’s match against Fulham.
Guardiola’s recovery will have been aided by the knowledge that City can prosper without him.
Although he did it from afar, it was a monumental victory for Guardiola, who now has 200 league wins with City in just 269 games, breaking Jose Mourinho’s previous Premier League record of 326 matches established in 2020.
For the first time since 2016, City have started a season with three straight league wins, sending a message to their title rivals that last season’s treble triumph has not satisfied their hunger for success.
When Julian Alvarez’s cross was stopped by John Egan’s outstretched arm in the 36th minute, City should have gone ahead.
However, Haaland smacked his penalty against the post. As the ball rebounded to safety, the Norwegian stood with a stunned expression on his face.
After scoring seven penalties in the Premier League, it was Haaland’s first miss.
When Haaland missed another fantastic chance from Kyle Walker’s cross, it appeared that the striker was having a bad day.
Wes Foderingham punched away a close-range chip from Haaland after Mateo Kovacic’s pass.
However, in the 63rd minute, Haaland rewarded City’s territorial control by rising above Jack Robinson at the far post to head his third goal of the season from Jack Grealish’s tempting cross.
City were in front before Blades replacement Bogle equalised in the 85th minute.
Walker was guilty of a wasteful backheel inside his own area while playing against the club he supported as a child, extending a United attack that culminated with Bogle’s ferocious strike bouncing in off Nathan Ake.
Rodri hammered into the top corner from 12 yards after Walker redeemed himself by recovering possession outside the United area, and City recovered the lead three minutes later.
(With inputs from AFP)
