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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.