The Canucks sent goalie Marek Mazanec to Tampa with a 2019 third-round draft pick and a 2020 conditional selection. If Vancouver is involved in the 2020 draft lottery, the pick will become a first-rounder for 2021.
Miller, 26, had 13 goals and 34 assists for the Lightning last season.
The native of East Palestine, Ohio, is known as a dynamic forward who can play all three positions.
"J.T. Miller is an experienced and versatile offensive contributor," Canucks general manager Jim Benning said in a statement. "This deal is about acquiring a player who can have an immediate impact in our top-six forward group for term."
Miller was originally selected 15th overall by the Rangers in 2011 and spent six seasons in New York before he was dealt to Tampa Bay in 2018.
The Canucks got Mazanec from the Nashville Predators in February but he did not play for Vancouver last season.
Later in the day, Vancouver traded forward Tom Pyatt and a sixth-round pick to the San Jose Sharks for Francis Perron and a seventh-round selection.
Both Perron and Pyatt spent last season in the American Hockey League.
The NHL draft kicked off in Vancouver on Friday night, with the New Jersey Devils taking American forward Jack Hughes first overall.
The Canucks took Russian forward Vasily Podkolzin with the 10th pick.