Miami obtained infielder Joe Panik and minor league reliever Andrew McInvale. The Blue Jays also received cash considerations from the Marlins.
The Blue Jays were targeting Cimber, a sidewinding right-hander, and took on Dickerson's remaining salary -- about $4.5 million -- to acquire him, a source told ESPN's Jeff Passan.
Blue Jays general manager Ross Atkins had said last week that one of his goals before the July 30 trade deadline was to add pieces to his bullpen.
"He is going to help our bullpen a lot,'' Blue Jays manager Charlie Montoyo said Tuesday before Toronto began a three-game series against the Mariners in Buffalo. "He gives us a different look. So I'm really looking forward to adding him to the group. And just like everybody else, he is going to get a chance to pitch in high leverage.''
Dickerson is sidelined with a bruised left foot and is expected to be in a walking boot for at least two more weeks. He was placed on the 10-day injured list by the Blue Jays on Tuesday. His departure ensures that 23-year-old outfield prospect Jesus Sanchez will get playing time for the Marlins in left field.
"We definitely look at this as a move about today and the future,'' Marlins general manager Kim Ng said Tuesday before Miami opened a three-game series in Philadelphia.
"We need to look at Jesus Sanchez as a piece of this. We got sped up in the process in terms of bringing Jesus up after the month he had in Triple-A. When we saw Jesus and saw how he's handling it up here, we felt more comfortable in trading Corey Dickerson. Jesus being able to take that spot at this point was part of the equation.''
Cimber, 30, is 1-2 with a 2.88 ERA in 34⅓ innings for the Marlins this season, his first with the team. He has struck out 21 batters and has allowed 11 walks. He has a 3.69 ERA in 170⅔ innings in 185 appearances in his career.
He is under the team control through the 2023 season.
Dickerson, 32, bats left-handed and is hitting .260 with two home runs and 14 RBIs in 62 games this season, his second with the Marlins. He is a career .282 hitter with 124 home runs and 401 RBIs in 890 games. He was an All-Star for the Tampa Bay Rays in 2017 when he hit a career-best 27 home runs.
Dickerson, who signed a two-year, $17.5 million contract with the Marlins in 2020, is scheduled to be a free agent after this season.
"He's going to be a good addition,'' Montoyo said. "We have a right-handed lineup, and to add him to the lineup, a left-handed bat like that, it's going to help us a lot.''
Montoyo is familiar with Dickerson from their time together in Tampa Bay, when Montoyo was a bench coach.
"He's a gamer,'' Montoyo said. "He's one of those guys that is always dirty because he plays the game the right way. He plays hard. It's all about getting him healthy now.''
Panik, 30, is hitting .246 with two home runs and 11 RBIs for the Blue Jays this season, his second with the franchise. He made starts at third base, second base and first base for Toronto this season. He is a career .268 hitter with 41 home runs and 251 RBIs in 765 games. He was an All-Star with the San Francisco Giants when he hit a career-best .312. He also won a Gold Glove at second base for the Giants in 2016.
He will be a free agent after this season.
"Joe is a guy who has experience, versatility,'' Ng said. "We think he'll be a great fit.''
McInvale, a 24-year-old right-hander, has a 2.55 ERA in 15 games at Class A and Double-A.
To clear room on their roster, the Blue Jays designated left-hander Travis Bergen for assignment.
The deal could be the first of several before the trade deadline for the Marlins, who are last in the NL East. Toronto is third in the AL East.
The Associated Press contributed to this report.