Kulikov, who assisted on Leopold's goal, restored the Panthers' one-goal lead when he scored 13:48 into the second on Florida's first shot against Mike Brodeur, who replaced Elliott moments earlier.
"It feels good for the back end," Leopold said. "Any way we can score goals and be able to contribute, it's big. I think everybody did a good job. Well done, all the way around."
Chris Phillips scored early in the second as each of the three goals in the game were produced on the power play by defensemen.
Ottawa came close to scoring on a goalmouth scramble with 17.3 seconds remaining but the referee ruled the puck hadn't crossed the line.
"Overall, I would have wished for a little better but we had a strong push at the end," Senators captain Daniel Alfredsson said. "I thought we had the goal, obviously, and we didn't get the call."
Leopold signed a one-year, $1.75-million contract with the Panthers on July 2, five days after he was acquired from Calgary in a draft-day trade for Jay Bouwmeester.
Alex Kovalev and Jonathan Cheechoo made their Ottawa debuts and both played a role in Phillips' tying goal 1:06 into the second.
Kovalev, who signed a two-year, $10-million deal with the Senators on July 6, began the scoring play with a pass to Mike Fisher. Cheechoo drew an assist when he got a touch on Fisher's centering feed as Phillips put the puck past Tomas Vokoun to tie it at 1.
"Kovalev made some really nice plays," Alfredsson said. "You can tell he's going to be valuable, especially on the power play, he draws a lot of attention. Jonathan had a couple of looks in the slot there on the power play but he didn't get a lot of chances because we couldn't really establish our cycling game."
Vokoun made 16 saves in two periods. Tyler Plante turned aside 10 shots in the third.
Elliott stopped 10 of 11 shots. Brodeur took over 12:53 into second and made nine saves.
The Senators obtained Cheechoo along with Milan Michalek from San Jose on Saturday in the trade that sent disgruntled All-Star Dany Heatley to the Sharks.
Michalek, who arrived in Ottawa on Tuesday, skated earlier in the day with other members of the Senators who did not face the Panthers.
Copyright by STATS LLC and The Associated Press
Arena: Canadian Tire Centre Location: Ottawa, Ontario
Attendance: 15,205 (82.2% full)
Jordan Leopold (power play) Assist: Michael Frolik, Dmitry Kulikov
Chris Phillips (power play) Assist: Jonathan Cheechoo, Mike Fisher
Dmitry Kulikov (power play) Assist: Bryan McCabe, Steven Reinprecht