Rodrigues went to the locker room after a collision with Carlsson on a power play 51 seconds into the third period, resulting in a five-minute major interference penalty and a game misconduct on Carlsson.
"I didn't think it was a major," Ducks coach Joel Quenneville said. "I think everybody could look at it a different way, but it looked like there was no intention there. He was just going through the middle. It was just one of those unfortunate hits."
Rodrigues returned just before Jacob Trouba scored on a rebound to extend it to 5-3 at 11:43 of the third period.
"We played well enough to be in it, but it’s not really well enough and up to our standards," Panthers forward Noah Gregor said. "So some stuff we got to work on.”
Chris Kreider scored off a tip to push it to 6-3 at 13:21, and Jansen Harkins swept the puck into the net while on his stomach at 14:47 for the 7-3 final.
The Ducks scored at least seven goals for an NHL-leading third time this season. They lead the League at 3.92 goals per game.
"We're super confident," Gauthier said. We have a really good team. A lot of depth, guys buying in in this locker room. There's no complaining. Coach ‘Q’ says stuff and we get it done."
NOTES: Gauthier (21 years, 289 days) became the fourth-youngest player in franchise history to record a hat trick for the Ducks, following Carlsson (18 years, 319 days on Nov. 10, 2023), Sam Steel (21 years, 51 days on March 26, 2019) and Isac Lundestrom (21 years, 115 days on March 1, 2021). ... Gauthier (seven goals, six assists) and Carlsson (three goals, nine assists) each extended his point streak to an NHL-career high seven games. They became the first set of teammates age 21 or younger to record simultaneous point streaks of at least seven games since Connor McDavid and Leon Draisaitl of the Edmonton Oilers in 2016-17. ... Marchand has 13 points (seven goals, six assists) in his first 12 games this season.