The manager was sacked after 10 games in charge of the League One side, who currently sit 22nd in the league table.
Winter said in a statement in the team's website: "First and foremost all at SUFC, from the board right the way through the club, wish David and [assistant] Lee [Carsley] all the best for the future.
"None of us wanted to be in this position at this point in the season. David's appointment was made as part of developing a new phase at the club.
"[Under-21 coach] Chris Morgan has been asked and agreed to take an interim role in charge of Sunday's team. He knows the players and, with Darren [Ward] and Travis [Binnion], will be doing his up most to get a return to winning ways."
The Blades face Coventry City on Sunday.