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.
Boston schools pull out free condoms over wrapping complaints
Millions of Getty images now available for free via embed tool