United did not disclose any contract details but British media reported that Ibrahimovic had signed a one-year deal to join up again with the Premier League club's new manager Jose Mourinho.
United have not paid a fee to sign the 34-year-old, who was out of contract after leaving Paris St Germain.
"I am absolutely delighted to be joining Manchester United and am looking forward to playing in the Premier League," he said in a club statement.
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"I have thoroughly enjoyed my career so far and have some great memories. I am now ready to create more special memories in England," he added.
"It goes without saying that I cannot wait to work with Jose Mourinho once again. He is a fantastic manager and I am ready for this new and exciting challenge."
Ibrahimovic has won titles in four different countries in a career which began with his hometown team Malmo in 1999 and has included spells at Ajax Amsterdam, Juventus, Inter Milan, where he played under Mourinho, Barcelona and AC Milan.
He has scored 392 goals in 677 games and won a trophy in each season since 2001.
"Zlatan needs no introduction. The statistics speak for themselves," Mourinho said. "Ibra is one of the best strikers in the world and a player who always gives 100 percent.
"He has won the most important league championships in the world of football, now he has the opportunity to play in the best league in the world and I know he will grasp this opportunity and will work hard to help the team win titles."
Ibrahimovic retired from internationals after Sweden's Euro 2016 exit last week, having won 116 caps and scored 62 goals.
He left PSG, having made 180 appearances and netting 156 goals, after breaking the Ligue 1 club's all-time league scoring record and winning a second straight domestic treble.
(Reporting by Ken Ferris and Martyn Herman, editing by Ed Osmond)
