The Rays also dealt minor league catcher Ronaldo Hernandez and minor league infielder Nick Sogard to the Boston Red Sox for right-handed pitcher Chris Mazza, left-handed pitcher Jeffrey Springs and cash.
Boston placed catcher Kevin Plawecki on the COVID-19-related injured list.
Curtiss went 3-0 with a 1.80 ERA in 17 games last year. He had a 6.30 ERA in 10 innings in the postseason for the American League champions.
Curtiss also has pitched for the Angels and Twins, and he has a 3.83 career ERA. He further strengthens the bullpen for the Marlins, who earlier added left-hander Ross Detwiler and right-handers Dylan Floro, Anthony Bass and Adam Cimber.
Mazza pitched nine games as a starter and reliever for Boston last year and went 1-2 with a 4.80 ERA. Springs went 0-2 with a 7.08 ERA in 16 games out of the bullpen.
Edwards was a fourth-round pick out of NC State in 2019. He hit .281 with nine homers in 73 games in the low minors that year.