The 34-year-old has had an outstanding finish to the season, scoring eight goals in seven games.
Cavani said: "I have developed a great affection for the club and everything that it represents.
"I feel a deep bond with my team-mates and the staff who work behind the scenes."
Cavani, who was undecided over whether to stay for a second year, added: "They give me extra motivation every day and I know that together we can achieve special things."
He has been one of the leading strikers in European football for the past decade, scoring a record 200 goals in 301 appearances for Paris St-Germain after joining from Napoli, where he made his name.
Cavani left the French club in June 2020 following a disagreement with Brazil forward Neymar.
Boss Ole Gunnar Solskjaer says he always wanted Cavani to stay for a second season.
"I said when Edinson signed that he would bring energy, power and leadership to this group and I haven't been proved wrong, he has been everything I thought he would be and more," said the Norwegian.
"As coaches we knew about his goalscoring record, however it is his personality that has brought so much to this squad, he has a winning mentality and has an unequivocal attitude to everything he does.
"Edinson is one of the last to leave the training ground and sets the tone for the upcoming youngsters with his approach to his profession every single day.
"I have always wanted him to stay and experience the way our fans will respond to him as a player, and this means he will hopefully get that opportunity."