The treble winners endured a difficult opening 45 minutes, culminating in a rare penalty miss from Erling Haaland, but the Golden Boot winner went from zero to hero to break the deadlock in the second half.
However, a Jayden Bogle leveller set up a nail-biting finale, where Rodri rifled a stunning strike into the top corner to maintain his side's unblemished record, as the Blades ended the chaotic contest with nothing to show for their praiseworthy efforts.
Citizens head coach Pep Guardiola - recuperating from back surgery - was not the only notable absentee for Man City at Bramall Lane, as Phil Foden, the star of the show against Newcastle United, found himself on the bench as Bernardo Silva returned to the fold.
City could have done with a dash of Foden's inventiveness in a frustrating first 45, as assistant manager Juanma Lillo witnessed his side pin the Blades back into their own half before struggling to play their way through the low block.
Paul Heckingbottom's injury crisis went from bad to worse in the 18th minute as Ben Osborn appeared to tweak his groin, and two minutes after Yasser Larouci's introduction, City had the ball in the net through Nathan Ake.
Aided by a slip from Wes Foderingham, the Dutchman volleyed home from close range after Rodri's header from a free kick, but the Spaniard was in an offside position as the ball was floated into the box and the goal was disallowed.
Another significant reprieve came Sheffield United's way in the 27th minute, as a deflected Haaland shot fell kindly into the path of Julian Alvarez, but Foderingham came charging out and made himself big to deny the Argentine from close range.
There was no respite for the Blades backline, though, and referee Jarred Gillett pointed to the spot in the 35th minute, as a sliding John Egan blocked Alvarez's cutback with his outstretched arm in the 35th minute.
As ever, Haaland would step up for his ostensible customary goal, but while he sent Foderingham the wrong way, the 23-year-old's penalty struck the post and rolled harmlessly away, sparking jubilant scenes behind the Sheffield United goal.
A couple more penalty shouts came and went for Man City and an aggrieved Haaland, who was on the receiving end of rough treatment from Sheffield United's centre-backs just before the break, although nothing that Gillett deemed spot-kick worthy.
Attacking in front of their own fans in the second half, Man City continued to pepper Foderingham's goal with shots from all angles, but Haaland and Rodri both missed gilt-edged chances before Foderingham stuck a strong hand up to deny the former's chipped attempt in the 61st minute.
Only a moment later, the beleaguered Blades goalkeeper thwarted Alvarez at his near post before Kyle Walker's long-range attempt flew over the top, but the champions' relentless pressure would belatedly pay off with 63 minutes gone.
Jack Grealish - who had seen plenty of the ball down the left-hand side - dinked a delicate cross to the back post, where Haaland was waiting to direct a bullet header into the back of the net from close range, although there was barely a hint of a celebration from the relieved Scandinavian.
Haaland made the net bulge once again eight minutes later, but the offside flag was inevitable, and Sheffield United refused to give up the ghost in the final 15 minutes, as Ruben Dias made a brilliant block from a Gustavo Hamer strike before Ederson's misplaced clearance fell to Anel Ahmedhodzic, but the centre-back's speculative attempt sailed harmlessly over.
Suddenly, there was a zest about Heckingbottom's team, and with 85 minutes on the clock, ex-Blades defender Walker gave his old employers the best gift he could - a horror mistake leading to the equaliser.
The Englishman's backheel inside his own box fell to Benie Traore, and the ball eventually found its way to Bogle, who had ample time to line up his shot and found the far corner via a slight deflection.
Bramall Lane was on cloud nine, but the hosts' euphoria lasted for just four minutes, as Walker - desperate to make amends for his howler - kept the ball in play on the right-hand side and sent in a low cross for Foden, whose touch let him down, but Rodri arrived to direct a thunderous half-volley into the top corner.
Seven minutes of added time was not enough for the Blades to respond again, as Man City returned to the summit ahead of next weekend's visit of Fulham, while Sheffield United now gear up for their EFL Cup second-round tie with Lincoln City on Wednesday.