Hudson had a career-high 12 strikeouts and took a six-hit shutout into the ninth, but Miguel Cabrera, Cody Ross and Aaron Boone singled to load the bases starting the inning. Wickman (1-1) came on, and Olivo hit his first pitch for a two-run double.
Joe Borchard was intentionally walked to reload the bases. Wickman fell behind Josh Willingham 3-0, then struck him out swinging.
With third baseman Chipper Jones playing in, Alfredo Amezaga flared an RBI single over him to tie the game.
On a 1-1 pitch to Dan Uggla, catcher Brian McCann let a low pitch from Wickman get past, and Olivo scored without a play.
The unearned run was the first run off Wickman this season, and the blown save was his first in seven chances.
Hudson came in a major league-best 0.62 ERA and dominated until the ninth.
His ERA rose to 1.22.
The Marlins' Scott Olsen pitched eight innings and struck out 10, but he gave up homers to three of the Braves' first seven batters - Kelly Johnson, Jones and Matt Diaz.
Henry Owens (1-0) pitched a 1-2-3 ninth.
Florida won two of three games in the series, and the Braves lost for the 12th time in their past 17 games Miami.
The Braves began the game tied for the NL lead 26 homers, and their first three batters hit the ball nearly 1,200 feet.
Johnson led off with his fifth home run, Edgar Renteria flied out to the warning track, and Jones hit his seventh homer.
Diaz ended an 0-for-17 skid with his second homer in the second.
In an attempt to change the Marlins' fortunes against Hudson, manager Fredi Gonzalez shook up the lineup, batting Hanley Ramirez third for the first time in his career. But Ramirez went 0-for-4 with two strikeouts, ending his 11-game hitting streak.