Half-time sub Oli McBurnie was the Blades' hero as he kept his cool to convert a penalty in the 13th minute of stoppage time, after being clattered by Hammers goalkeeper Alphonse Areola four minutes or so earlier.
Earlier Ben Brereton Diaz had cancelled out Maxwel Cornet's opener with a goal on his Premier League debut but James Ward-Prowse looked to have won it for West Ham with a 79th-minute penalty, after Gus Hamer brought him down. Blades sub Rhian Brewster saw red for a out-of-control challenge on Emerson Palmieri but United stuck to their task and got their reward deep into stoppage time.
Here's how we rated the Blades' players on a topsy-turvy afternoon...
1. Wes Foderingham 6
Had no chance with West Ham's opener and almost the same odds from their winner, facing dead-ball expert Ward-Prowse after Hamer had fouled Ings. Otherwise he was solid with the little he had to do, collecting an Ings shot nearly down to his left and excelling early on to get a hand on Ings' attempted lob that otherwise could have landed in the back of his net | Getty Images
2. Jayden Bogle 6
Came back into the side as one of five changes from Gillingham but was caught ball-watching for the visitors' opener as Ings' shot was deflected into the path of Cornet at the back post, who had the freedom of Bramall Lane to pick his spot and fire past Foderingham. Then began his redemption with a good block to prevent Ings potentially doubling the Hammers' lead, and his calmness and composure on the ball kickstarted the move that led to Brereton Diaz's equaliser. Then good defending and a big tackle prevented a dangerous Hammers break | Getty Images
3. Anel Ahmedhodzic 6
The subject of intense transfer speculation over the last week, the Bosnian had to show it had not affected him but he hardly covered himself in glory for the Hammers' opener either, failing to react to the deflection or realise Bogle had moved too narrow and left Cornet in acres of space. Attracted the frustration of fans with a backwards pass late on when the Blades were chasing a route back into the game | Getty Images
4. Jack Robinson 5
A typically calm but committed performance from Robinson for the most part but he may look back on what looked like being West Ham's winner with some regret, Ings being fouled by Hamer for the penalty after the forward had burst through Robinson's tackle. For a man who normally takes ball and a fair bit of man it was uncharacteristic and his only real misstep all afternoon but it almost proved pivotal | Getty Images