Smith (1-0) picked up the win Thursday, striking out two in a perfect 10th inning during the Marlins' victory over the Rockies.
The Marlins acquired Smith from the Rays on Wednesday in exchange for cash considerations and selected Smith's contract from Triple-A Jacksonville, Marc Topkin of the Tampa Bay Times reports.
Burch signed a minor-league contract with the Rays on Jan. 2.
Smith agreed to a contract with the Hanwha Eagles of the Korean Baseball Organization on Sunday, Dan Kurtz of MyKBO.net reports.
Smith cleared waivers Thursday and was outrighted to Triple-A Las Vegas, Matt Kawahara of the San Francisco Chronicle reports.
Smith was designated for assignment by the Athletics on Tuesday.
Smith allowed three earned runs on five hits over one inning in a loss to the Yankees on Friday.
Smith, who owns a 1-1 record, 5.94 ERA and 1.34 WHIP across 24.2 innings, has logged just two appearances since June 26 as the second half of the season begins.
Smith allowed four earned runs on three hits and a walk over one inning in a win over the Angels on Monday. He also hit a batter and recorded a strikeout.
Smith fired a scoreless seventh inning in a loss to the Angels on Sunday, allowing a hit and a walk.