The Old Lady have also sealed a move for Milan right-back Mattia De Sciglio, who will join for £11m.
Szczesny, 27, spent the past two seasons on loan at Roma after losing his starting berth at Emirates Stadium and will travel to the United States with his new team-mates on Thursday for their pre-season tour.
The goalkeeper would have been playing second fiddle behind Petr Cech if he were to return to Arsenal and therefore set his sights on a transfer to Turin where he will be earning a cool £3.5m-a-year.
De Sciglio, meanwhile, will reunite with his former boss Massimiliano Allegri, who gave the Italy international his first taste of senior club football in 2011 with Milan.
Juve coach Allegri sees Szczesny as the long-term successor of Gianluigi Buffon, who is expected to hang up his gloves in 2018.
Szczesny's permanent exit paves the way for Emiliano Martinez to become Cech's deputy in 2017-18, with Arsenal ready to cash in on wantaway David Ospina, too.
The Colombian shot stopper is keen to leave the club in order to get regular first-team action elsewhere and an offer in the region of £5m would suffice to lure him away from the Gunners.
There was interest from Fenerbahce earlier in the summer, but a move to the Istanbul giants eventually fell through after the Turkish club refused to increase their £3m offer for the goalkeeper.