The Nationals also will get a player to be named and will send cash to the Blue Jays.
Storen, 28, had sought a deal, having lost his closer role in late July when the Nationals acquired Jonathan Papelbon from Philadelphia.
Storen had converted 29 save chances and had a 1.73 ERA when the Nats got Papelbon. Shifted to an eighth-inning role following that deal, Storen struggled and finished the season with a 3.44 ERA without another save. His season ended in mid-September when he slammed his locker in frustration and broke his right thumb.
He is 21-13 with 95 saves and a 3.02 ERA in six major league seasons, all with the Nationals. He had a 1.12 ERA in 65 games in 2014.
For all his success in the Washington bullpen, Storen also was tagged with two big blown saves in the playoffs.
Storen failed to close out St. Louis in the ninth inning of the deciding Game 5 in the 2012 NL Division Series, and he let a ninth-inning lead slip away in Game 2 of the 2014 NLDS against San Francisco.
He took to Twitter to express his excitement about joining the Jays as well as thanking the Nationals' fans.
Thanks DC for all the memories. Thanks to the fans, you were with me through thick & thin. I'm happy to have been part of something special— Drew Storen (@DrewStoren) January 9, 2016
Storen joins a Toronto relief corps that relied on Roberto Osuna as the main closer last year. At 20, Osuna led the team with 20 saves.
Revere, 27, figures to take over the center field and leadoff job that Denard Span held. Span signed with San Francisco this week as a free agent.
Revere has hit better than .300 in each of the past three seasons. He batted a combined .306 for the Phillies and Toronto last year, with a .342 on-base percentage and 31 stolen bases.
The Nationals had hoped to boost their offense from the left side of the plate in the offseason. Coming off a disappointing second-place finish in the NL East, they recently signed free-agent second baseman Daniel Murphy; Murphy and Revere both bat left-handed.
Revere led the NL with 184 hits in 2014 with the Phillies. In six seasons with Minnesota, Philadelphia and Toronto, he has hit .295 with 176 steals.