The Senators held a 2-1 lead with time running out in the second period, but Aleksander Barkow tied it on a deflection with 2:49 left in the stanza.
After Gilbert's power-play goal gave the Panthers the lead, Tomas Kopecky clinched the win with a shorthanded tally that came with 1:41 remaining.
Florida has won five in a row, its longest winning streak of the season. Scott Clemmensen earned the win with 32 saves after giving up a pair of first-period goals.
Craig Anderson suffered the loss while stopping 31 shots. It was Ottawa's fourth loss in six games.