A 39-year-old right-hander, Grilli is 1-2 with a 5.29 ERA and two saves this season in his comeback from a career-threatening injury. Grilli's 2015 season ended when he tore his left Achilles tendon on July 11. He had 24 saves and a 2.94 ERA in 36 games before the injury.
Toronto general manager Ross Atkins said Grilli is expected to join the Blue Jays on Wednesday. He said Grilli is "someone that we've liked for a while."
"There's a lot that we like about Jason and how he complements this team," Atkins said. "The first thing that comes to mind is experience. He's been on winning teams in very high-leverage situations and has got a lot of swing and miss in his career. Those are good attributes."
Grilli's father, Steve Grilli, was a pitcher who made a brief appearance with the Blue Jays in 1979.
Arodys Vizcaino has six saves and a 1.66 ERA in his emergence as the Braves' new closer.
Atlanta is rebuilding and entered Tuesday at 15-35, tied with Minnesota for the worst record in the majors. General manager John Coppolella said the trade helps open a spot for another young arm.
"We wanted to go with young pitching," Coppolella said. "What we've tried to do in a year like this is to try to find out what we have with the arms we have here."
The Braves expect to have right-hander Jim Johnson, who is out with a right groin strain, come off the 15-day disabled list Friday. Coppolella said right-hander Shae Simmons, recovering from Tommy John surgery, also should soon join Atlanta's roster.
Toronto also gets $2,175,000 from Atlanta, offsetting much of the remaining $2,371,585 from Grilli's $3.5 million salary this year. His deal includes a $3 million team option for 2017 with a $250,000 buyout.
Atlanta obtained right-hander Sean Ratclilffe, an 18th-round pick in 2013 who was 2-2 with a 3.60 ERA in 22 games with Class A Vancouver last year. The Braves recalled right-hander Ryan Weber from Triple-A Gwinnett to fill Grilli's spot on the active roster.
Ratcliffe, 21, was in the Blue Jays' extended spring training and will report to the Braves' extended spring program.
"We had seen him two weeks back, and our scouts who saw him thought he could help the Braves organization," Coppolella said. "We'd like to see him throw for us first and then go from there and see where he can help us."
As rumors of the impending trade spread Tuesday, Grilli went through his normal pregame routine, including shagging fly balls in the outfield during batting practice. The trade became official about 30 minutes before the Braves' game against the San Francisco Giants.
Grilli has 76 career saves, including 33 with Pittsburgh in 2013. He made his debut in the majors with Florida in 2001 and has also pitched for the White Sox, Detroit, Colorado, Texas and the Angels.
Grilli thanked Braves fans on his Twitter account on Tuesday, posting, "From the bottom of my heart...thank you."