The 33-year-old striker turned the ball home from close range in the 88th minute after Chris Mepham had earlier given the Cherries the lead against the run of play.
The Blades almost scored with their first attack of the game, Bournemouth goalkeeper Aaron Ramsdale making a smart save to keep out David McGoldricks shot from the edge of the box.
United, playing in the Premier League for the first time since 2007, created another good opening in the 21st minute, but Ramsdale got down well to save Callum Robinsons 25-yard drive before John Lundstram dragged his shot wide.
Eddie Howes team had most of the first-half possession but were unable to fashion a clear-cut chance, as the teams went into the interval all square.
United continued to create the better openings at the start of the second half, with Mepham having to make a great last-ditch challenge to stop Republic of Ireland international Enda Stevens breaking free, while McGoldrick was only just off target with a 25-yard shot.
Bournemouth went close to breaking the deadlock with their first clear opening in the 62nd minute, Blades goalkeeper Dean Henderson making an outstanding stop to somehow keep Josh Kings 25-yard strike out.
Buoyed by the near-miss, the Cherries opened the scoring 60 seconds later, Mepham turning the rebound home after Henderson had only managed to parry Callum Wilsons weak shot.
After playing so well for over an hour, United were rocked by falling behind and looked set to be slipping to an unfortunate defeat until Sharps late intervention.