Minor, the front-runner for the fifth slot, effectively pounded the strike zone. Of his 48 pitches, 35 were strikes.
"I felt pretty good today," Minor said. "Not just because of the scoreboard, but I think that was my best outing, just because of the overall pitch sequences, locating strikes and working both sides of the plate."
Miami's Carlos Zambrano, meanwhile, also had an impressive four-inning stint. The veteran right-hander, who will be either the Marlins' fourth or fifth starter, was perfect through three innings, with four strikeouts.
The Braves opened the scoring in the fourth, when they were able to scratch out a run by capitalizing on a leadoff walk to Michael Bourn. Martin Prado singled and Brian McCann delivered an RBI single down to right.
The Marlins rallied in the seventh, sending eight men to the plate and pushing across a pair of runs on RBI doubles by Logan Morrison and John Buck. But Atlanta tied the game in the ninth when catcher Christian Bethancourt hit his first homer of the spring off Miami reliever Evan Reed. The clubs played the 10th inning before calling it a day.
Overall, it was a solid four-inning outing for Zambrano, who struck out five while giving up a run on two hits and one walk. The right-hander threw 49 pitches, with 29 strikes.
"Much better, compared to the last time, it was good," Zambrano said. "All of my pitches. I thank God that I'm back with kind of the form I had in the days when I used to be really good. I was very pleased with the outing."
For the second straight day, Marlins manager Ozzie Guillen found himself in a dispute with the umpires over a ball ruled foul down the first-base line.
With a runner on first in the third inning, Scott Cousins chopped a grounder that deflected off first baseman Freddie Freeman. Home-plate umpire Ed Rapuano pointed fair, but first-base umpire CB Bucknor signaled foul. Guillen rushed to the umpires, who explained their decision.
On Monday in Fort Myers, Fla., Guillen was ejected for arguing a similar close-call grounder along the first-base line.
Miami's Heath Bell threw a perfect fifth, with a strikeout.
Atlanta's Kris Medlen tossed two scoreless innings, scattering two singles.
Up next: After two consecutive games in Jupiter, the Braves will finally head back to Lake Buena Vista, Fla., on Wednesday to face the Washington Nationals at 6:05 p.m. ET, in a game that will be broadcast on MLB.TV. Right-hander Julio Teheran, who has allowed eight runs in five innings, will start for Atlanta. On the other side, Stephen Strasburg will be on the mound for the Nationals.