The Blades looked to have earned their first point of the season when Jayden Bogle came off the bench to equalise in the 85th minute, after Erling Haaland had earlier put the visitors ahead.
But poor defending on the wing gave City one more chance and Rodri, who scored the winner in the Champions League final, repeated the trick at Bramall Lane with a superb finish to earn all three points for the reigning champions just three minutes later.
Earlier Haaland missed a penalty and City had two goals ruled out for offside. Here are our player ratings from a game that left United with no points, but a lot of pride...
1. Wes Foderingham 8.5
Outstanding. An early slip was his only real mis-step, Ake poking home before the offside flag came. Made an outstanding close-range save to deny Alvarez what looked a certain opener and then excelled again to keep out Haaland’s chipped effort. Had no chance when Haaland eventually broke the deadlock or when Rodri won it with a superb finish. Didn’t deserve to lose | Sportimage
2. George Baldock 7
Had another great running battle with Grealish and brought a great cheer to Bramall Lane when he caught his old pal with a late challenge, picking up a deserved booking in the process. Couldn’t prevent the cross for Haaland’s opener but there’s not much shame in that considering the calibre of player he was up against. Made way for Bogle | Getty Images
3. Anel Ahmedhodzic 7
Had United fans interested for a moment as he tried an ambitious volley after Ederson’s poor clearance, that didn’t end up a million miles away. Was spoken to by ref Gillett early in the game with Haaland as they enjoyed a good physical battle, and doubled up a lot with Baldock to try and snuff out the threat of Grealish cutting in from the left. Could have scored another dramatic equaliser when he threw himself as McBurnie’s cross but just couldn’t get enough on it | Getty Images
4. John Egan 7
On first glance Egan looked harshly penalised when conceding a penalty but replays showed it to be a fair decision, with his hand in what’s deemed an ‘unnatural position’ these days as Alvarez crossed from the left. Otherwise marshalled City’s numerous threats well enough for the most part. Booked late on, either for his ‘foul’ on Grealish or his reaction - or a combination of both | AFP via Getty Images