Bradley, who is ranked the club's No. 2 prospect by MLB.com, had a home run and an RBI double. Mike Carp added a solo homer in the fourth and Will Middlebrooks connected on a two-run shot in the seventh.
Miami pushed across a run in the first inning, but the Boston lefty settled down nicely and finished with an impressive five-inning, one-run outing with four strikeouts.
After issuing a one-out double to Casey Kotchman in the second inning, Lester, who threw 54 pitches (38 strikes), retired the final 11 batters he faced.
Marlins non-roster invitee Kevin Slowey had a mixed results afternoon. He struck out five in three innings, but the right-hander gave up three runs on two hits with two walks and a hit batter.
Slowey is competing for the fifth starter or long reliever role.
Bradley jumped all over Slowey's first pitch of the game and drove an opposite-field homer to left.
Slowey regrouped and struck out four of the next six he faced.
The Marlins tied it at 1 in the first. Juan Pierre singled and scored on Chris Valaika's RBI single to right.
"Anytime you get somebody like Juan on first two pitches and haven't really gotten your rhythm yet, it makes it a little harder to get that ball back down in the zone," said Lester. "But after the first, I felt good. I mixed some pitches in, mixed some offspeed pitches in, which was nice, especially after my last start, [when I wasn't] able to get those over for strikes. All in all, felt pretty good."
In the third inning, Boston regained the lead on Jarrod Saltalamacchia's two-run double. The Red Sox received a scare in the inning when Mike Napoli was plunked on the left wrist by a Slowey fastball. Napoli was tended to by the trainer, but he remained in the game.
Miami turned to Tom Koehler in the fourth inning, and the right-hander was greeted much the same way as Slowey. Like Slowey, Koehler surrendered a homer to the first batter he faced.
Carp connected on a home run to left to open a two-run fourth. Bradley added an RBI double in the inning.
Up next: Right-hander Clay Buchholz makes his third start of Spring Training in Tuesday's 1:35 p.m. ET home game against the Blue Jays. Top prospect Allen Webster, who has looked dominant in camp, will work out of the bullpen. Lefty Andrew Miller and righty Andrew Bailey are also slated to pitch. Position players Jacoby Ellsbury, Dustin Pedroia and Napoli are all expected to be in the lineup.