Glasgow followed up the signing of Tonga forward Viliami Ma'afu with a move for Newcastle's 2010-11 player of the season.
The second-rower, who has made 82 appearances since making his Falcons debut in 2007, has signed a deal until May 2014.
Swinson, 25, is eligible to play for Scotland, courtesy of a maternal grandmother from Glasgow.
Swinson told www.glasgowwarriors.org: "I'm really pleased to have signed for a club with the ambition and bright future of Glasgow Warriors.
"They've made real progress in the RaboDirect PRO12 this season, and with the signings and re-signings they've made, they're clearly very determined to build on that and achieve success.
"I'm thoroughly looking forward to throwing myself into the cause come the summer, but for now it's about focusing on helping Newcastle Falcons secure their Aviva Premiership status.
"There is a great team environment at the club, and everything I've seen and heard about the Warriors suggests that's very much the case in Glasgow as well.
"It's going to be a really positive place and team to be part of. I leave the Falcons on good terms and the move has been handled well on both sides."
Head coach Sean Lineen, who has also secured the return of Sean Lamont for next season, added: "Tim is a player who has consistently stood out in one of the toughest testing grounds for a second-row forward, the English Premiership.
"He's a battle-hardened competitor who relishes the physical side of the game, and he's not one for taking a backward step when the pressure's really on."
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