The Revs announced in February a deal had been agreed with Arsenal for Turner, subject to the completion of a medical.
The 28 year-old now heads to Emirates Stadium for a reported £7.5million ($9.2m) transfer fee, boosting Mikel Arteta's goalkeeping options.
Turner is likely to serve as back-up to Aaron Ramsdale, with Bernd Leno – the number one prior to the England international's arrival last season – linked with a move away from the London club.
The USMNT star arrives in England after an outstanding 2021 season, in which he was the MLS Goalkeeper of the Year and made the MLS Best XI as the Revs won the Supporters' Shield.
Turner also appeared in the MLS All-Star Game for the first time and was named MVP after saving two penalties in a shoot-out win over a Liga MX outfit.
"Matt is an experienced goalkeeper who will bring high quality to our squad," Arteta said. "He has shown with his performances in recent years in both MLS and at international level with the United States that he is a goalkeeper who will give us added strength going into next season.
"We welcome Matt and his family to the club and we look forward to working together in the years to come."
Turner follows Marquinhos and Fabio Vieira to Arsenal in this transfer window and appears set to be joined by Gabriel Jesus, for whom the Gunners have reportedly agreed a deal with Manchester City.