Gunners boss Arsene Wenger told his club's website: "We have a strong squad but a young squad.
"Mikael's versatility, experience and calibre will provide the extra depth we need to reinforce our challenge for honours this season.
"His defensive adaptability will serve us well, and it's a big plus that Mikael has top-level experience and a great understanding of football in the Premier League."
Silvestre added: "I've spent a large part of my career at United and I'll remember that time fondly but feel I'm ready for a new challenge.
"Arsenal are known for playing technical and skilful football, and have a young but experienced squad.
"I'm really excited about this next stage of my career, hopefully playing my part in helping the club to new successes."
Silvestre had faced competition for the left-back slot at Old Trafford since Patrice Evra arrived from Monaco in 2006, while Nemanja Vidic has established himself as Rio Ferdinand's first-choice central defensive partner.
He has become the first player to be sold to the Gunners by United since Brian Kidd in 1974.
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