The City midfielder crashed home a shot from inside the area less than three minutes after Blades substitute Jayden Bogle had cancelled out Erling Haaland’s second-half header with an 85th-minute equaliser.
It was a breathless finish to a game City had dominated – Haaland missed a first-half penalty – but Rodri’s strike finally ended the Blades’ brave resistance and sealed absent manager Pep Guardiola’s 200th win in the English top flight.
With City assistant coach Juanma Lillo in charge of the visitors’ dugout as Guardiola recovers from back surgery in Barcelona, it took 12 minutes for his side to carve out their first chance as Haaland headed straight at Blades goalkeeper Wes Foderingham.
Foderingham denied the visitors again in the 26th minute by deflecting Alvarez’s shot for a corner.
City were gifted the chance to take a first-half lead when Alvarez’s cross struck John Egan on the arm as the Blades skipper slid in to block and referee Jarred Gillett pointed straight to the spot.
But Haaland failed to convert as his low, left-footed penalty struck Foderingham’s left-hand post.
Blades defender George Baldock was booked for his juddering challenge on Jack Grealish before the interval and the home fans gave their side a standing ovation at the half-time whistle with the score at 0-0.
Lillo stepped out into the technical area for the first time shortly after the restart to see Haaland shank Kyle Walker’s cross inches wide.
There was now more urgency to City’s approach play. Rodri pulled a low effort wide, Foderingham rescued the Blades again by palming Haaland’s dinked effort for a corner and Alvarez lashed Grealish’s lay-off just wide.
Walker then blazed over before City made the breakthrough in the 63rd minute.
Grealish was marshalled by Baldock on the left edge of the area, but made space to clip the ball to the far post and Haaland powered home his third league goal of the season with a towering header.
The Blades responded as substitute Oli McBurnie headed wide from a corner and at the other end Foderingham kept out Mateo Kovacic’s free-kick.
Bramall Lane erupted in the 85th minute when substitute Bogle arrowed a shot inside the far post to haul his side level following an earlier error by former Blade Walker.
But before a crowd of 31,336 could catch breath, Walker made amends by out-muscling Larouci out wide and his cross was lashed home by Rodri after Phil Foden had mis-controlled.
The Blades almost snatched another equaliser when Anel Ahmedhodzic just failed to latch on to McBurnie’s cross, but City held firm to maintain their 100 per cent start.