Rodri strikes late as Man City extend perfect start
Haaland drove a first half spot-kick against the post before his header put City in front after the break at Bramall Lane.
City were in danger of squandering the points when Jayden Bogle equalised against the run of play with just five minutes left.
But Rodri saved City with a blistering strike three minutes later as Pep Guardiola's side moved to the top of the table after a third successive victory.
City's latest success was achieved without Guardiola, who was watching on television while he recuperates after back surgery in Barcelona.
The City boss was in constant contact with his assistant Juanma Lillo, who took the reins at Bramall Lane and will also be in charge for the champions' clash with Fulham on Saturday.
Guardiola's rehabilitation will have been aided by the knowledge that City can thrive in his absence.
Although he achieved it from afar, it was a landmark victory for Guardiola, who now has 200 league wins with City in just 269 games, beating the previous Premier League record of 326 matches to reach that milestone set by Jose Mourinho in 2020.
For the first time since 2016, City have opened a season with three consecutive league wins in an ominous statement to their title rivals that last season's treble triumph has not sated their appetite for success.
City should have gone ahead in the 36th minute when Julian Alvarez's cross was blocked by John Egan's out-stretched arm.
But Haaland hammered his penalty against the post. The Norwegian stood with a look of disbelief etched on his face as the ball rebounded to safety.
It was Haaland's first penalty miss in the Premier League after scoring his previous seven.
When Haaland steered another good chance wide from Kyle Walker's cross, it looked like a day to forget for the striker.
Rodri to the rescue
Haaland was denied again from Mateo Kovacic's pass, his close-range chip clawed away by Wes Foderingham.
But Haaland finally rewarded City's territorial dominance in the 63rd minute, rising above Jack Robinson at the far post to head his third goal this season from Jack Grealish's teasing cross.
City were cruising but Blades substitute Bogle snatched a shock equaliser in the 85th minute.
Playing against the club he supported as a boy, Walker was guilty of a needless backheel inside his own area, prolonging a United attack that climaxed with Bogle's fierce shot deflecting in off Nathan Ake.
City responded superbly and regained the lead three minutes when Rodri lashed into the top corner from 12 yards after Walker redeemed himself by winning back possession outside the United area.
Aston Villa powered to a 3-1 win at Burnley as Matty Cash's double extended a remarkable goal spree from Unai Emery's side.
After losing 5-1 at Newcastle in their season opener, Villa had rebounded by routing Everton 4-0 and crushing Hibernian 5-0.
They made it 12 goals in three games.
Burnley have lost their first two games this term.
John McGinn's eighth minute pass found Ollie Watkins and his cross was perfectly weighted for Cash to finish with aplomb from close-range.
Cash netted again in the 20th minute, smashing home after a swift exchange of passes with Moussa Diaby.
In the 47th minute, Burnley's Lyle Foster fired home to become the first South African to score a Premier League goal since Steven Pienaar netted for Everton in 2013.
But Diaby ended Burnley's uprising in the 61st minute with a clinical finish from Lucas Digne's cut back.
Later on Sunday, Newcastle and Liverpool meet at St James' Park in an intriguing clash between two sides with title ambitions.