Francoeur connected on his two-run shot to left off Justin Nicolino with one out in the fourth, putting the Braves ahead, 3-1. Hector Olivera doubled to open the frame.
"I think Nicolino's ball was probably wind blown," Marlins manager Don Mattingly said. "I think we kind of know what Nico is, a guy who is going to change speeds. He's going to move the ball in and out. Not going to be an overpowering guy, but a guy who knows what he's doing. He does a lot of things well. I thought he was OK."
The Marlins closed the gap to 3-2 on Christian Yelich's RBI double, which scored prospect J.T. Riddle, who pinch-ran for Miguel Rojas, who exited after his second single of the afternoon.
Norris, who has locked up a rotation spot and threw two perfect frames in his first Grapefruit League start, set down the first five batters he faced on Thursday before Jeff Mathis doubled to left.
Miami had three hits in the third and the lone run Norris gave up came on Rojas' two-out RBI single to center. Norris picked off Ichiro Suzuki at first with runners on the corners and one out. But Rojas was able to pick up the run with his single to center.
Don Kelly led off the third with a bunt single to the left side after the Braves put on a dramatic shift for the left-handed hitter.
"The more we do it in Spring Training, the more we will get people in positions," Braves manager Fredi Gonzalez said. "But we're going to do a little more shifting. We have some numbers [on Kelly], and the numbers say he bunts rarely."
Marlins right-hander Jose Urena, who likely will open the season in Triple-A New Orleans' rotation, logged 2 2/3 innings, permitting a run on three hits with a strikeout and a hit batter.
Urena ran his fastball up to 97 mph, and after throwing mostly fastballs to start the game, mixed his pitches better after allowing a run in the first.
The Braves struck in that inning on Adonis Garcia's two-out RBI double to left. Ender Inciarte, who led the game off with a single to center, scored from first. Emilio Bonifacio and Chase d'Arnaud added insurance runs in the ninth with an RBI triple and RBI double, respectively.
One of the more eventful innings came in the top of the eighth when Miami reliever Kyle Barraclough struck out three, but also walked two, including top prospect Dansby Swanson. The Braves had runners on first and second when Barraclough, who topped out at 96 mph, struck out Mallex Smith on an 82 mph slider.
Up next for the Braves: When Atlanta plays its first split-squad games of the year on Friday, it will get a look at the two veteran non-roster invitees who are fighting for a rotation spot. Kyle Kendrick will start against his former team, the Phillies, at Disney, while Jhoulys Chacin goes against the Cardinals in Jupiter. Freddie Freeman, Nick Markakis and A.J. Pierzynski are scheduled to play against the Phillies. Ender Inciarte will be back in the leadoff spot for the contest against the Cardinals, which will be available on MLB.TV. First pitch for both games is scheduled for 1:05 p.m. ET.
Up next for the Marlins: The Marlins will face the Twins in Fort Myers at 1:05 p.m. ET on Friday. Jarred Cosart will make his second Grapefruit League appearance and first start. The right-hander is competing for a rotation spot. He was shaken up in his first outing, when he bruised his right knee on a collision while covering third base. Dee Gordon, whose brother Nick is in the Twins' system, will play, as will shortstop Adeiny Hechavarria and outfielder Marcell Ozuna.