The Swans had threatened a winner when Barrie McKay rattled his volley off the crossbar.
However, their comeback was capped when debutant Yan Dhanda slotted Jefferson Montero's shot into the goal after coming on as a substitute.
Oli McBurnie was able to equalise for United, as he smashed in the rebound of Dean Henderson's save in minute 71.
The Blades had led until 62 minutes, when George Baldock was able to swipe home Enda Stevens' cross.
Sheffield had begun as the brighter of the two sides, and should have taken the lead early on as Lee Evans' shot went just wide.
Swansea started cautiously, and rarely threatened, but picked up the tempo after half-time.
McBurnie was difficult for Swansea to contain, and went close on a few occasions.
The introduction of substitute Montero was the real difference, as he was able to terrorize the Blades' defence before he teed up the winner.
His cross found McKay who, despite being fouled, was able to slip the ball through to Dhanda who slotted home for the perfect start to Championship football.