The Lightning received the rights to forward Jeff Costello and a seventh-round pick in the 2015 NHL Draft in exchange for their second-round pick (No. 50) in the 2014 draft.
Garrison, 29, had seven goals and 33 points in 81 games this season, his second in Vancouver after signing a six-year contract. Garrison, who waived his no-move clause, has four years remaining on a contract that has an average annual value of $4.6 million.
"When it all kind of happened I was shocked and I didn't want to go anywhere else. I only signed here two years ago and wanted to be here the whole time for sure," Garrison told Team 1040 AM in Vancouver.
The trade was one of three made by the Canucks in the hours before the start of the NHL Draft. Vancouver also dealt center Ryan Kesler to the Anaheim Ducks in a multi-player deal and acquired forward Derek Dorsett from the New York Rangers for a third-round draft pick acquired in the Kesler trade.
One of the players acquired in the Kesler deal, defenseman Luca Sbisa, figures to step in to Garrison's slot on the blue line.
"The Jason move was we're trying to get a little bit younger on the back end, so Luca gives us youth on the back end," Canucks general manager Jim Benning said on a media conference call. "And going into free agency we wanted to free up some money to be in the game on some of these free agents come July 1."
Benning said the Garrison and Kesler trades didn't mean the Canucks were entering a rebuild.
"We acquired players that we thought are going to have an impact on us winning next year," he said. "We've been talking about bringing in some good energy guys that are going to work hard and compete hard physically and he's going to do that for us. So that's kind of why we did it, to help our team. If it was going to be a rebuild then maybe we wouldn't have tried to fight so hard to get players that can come in and help our team win next year."
Garrison entered the NHL as an undrafted free agent and had a career-high 16 goals with the Florida Panthers during the 2011-12 season. He has 38 goals and 108 points in 318 games with the Panthers and Canucks.
"They were a very good team," Garrison said told Team 1040 AM when asked about playing against the Lightning last season. "Young, skilled and energetic, and I think they killed us to be honest.
"This team has a lot of big pieces and they're going in the right direction. I'm hoping I can fit in and be a part of that."
Costello, 23, was a fifth-round pick (No. 146) of the Ottawa Senators in 2009. He played four seasons at the University of Notre Dame, scoring 13 goals and 22 points in 40 games last season.