Derek MacKenzie, Jonathan Huberdeau, Brandon Pirri and Corban Knight also scored for Atlantic Division-leading Florida in the opener of a season-high six-game road trip. Roberto Luongo made 31 saves to improve his career record against Buffalo to 18-7.
Ryan O’Reilly scored for the Sabres, who have lost five straight games in regulation and fell to 0-4-1 in their last five home games.
Huberdeau’s fantastic breakaway goal made it 4-1, his initial deke moving Ullmark well out of position before tucking the puck into the open net with 3:54 to go.
Pirri added an empty-net goal with 1:21 remaining to round out the scoring.
Eichel caused a turnover early in the first, only to see Luongo stop both his initial shot and rebound attempt.
The Panthers opened the scoring when Huberdeau skated the puck across the Sabres' zone before sliding it across to Jagr. The 43-year-old fired his team-best 14th goal of the season past Linus Ullmark to make it 1-0.
Buffalo tied it on a three-way passing play early in the second period. Jamie McGinn carried the puck down the left wall, sliding it to Eichel. Parked to the right of Luongo, Eichel fired a no-look backhand pass to the backdoor, where O'Reilly fired a hard shot past Luongo.
Another power-play chance for the Sabres yielded a short-handed breakaway for Aleksander Barkov, but Ullmark kept his position to deny the Panthers' centre.
MacKenzie put the Panthers ahead for good at the 8:54 mark of the second, when he raced up the right side of the ice and, given time to size up Ullmark, wristed a shot over the goalie's glove.
The Sabres had an advantage in shots on goal for the majority of the game, but Luongo's goaltending combined with an opportunistic Panthers attack to control the play.
That's how Florida took a 3-1 lead early in the third, when Knight was the beneficiary of a puck that bounced oddly into the slot.
Sam Reinhart did force Luongo into a pair of tricky saves to bookend the Panthers' third goal.
Jagr had a chance on a 2-on-1 with Barkov with just over 5 minutes to play, but Ullmark's save came with a penalty to Jake McCabe that sent Buffalo onto the penalty kill.
NOTES: Buffalo scratched defenceman Mike Weber in favour of Carlo Colaicovo, who had not played since Dec. 7. ... The Sabres start a three-game road swing on Friday at Chicago. ... Florida next visits Ottawa on Thursday.