Willingham hit a two-run double in the top of the first inning to start the scoring. Jeremy Hermida hit a sacrifice fly and then Boone hit a two-run homer in the third to up Florida's lead to 5-0.
Amezaga had three hits and Boone had two for the Marlins, who have won three of their last four games.
Kelly Johnson added a two-run single and Scott Thorman had three hits for the Braves, who have dropped four of five.
Loser Kyle Davies (3-4) yielded five earned runs on six hits and two walks in five innings. He struck out six.