"We showed a lot of character and we battled hard," Vokoun said. "We had good fortune, but sometimes that's what you need."
Vokoun made a glove save on a short-handed penalty shot by Tampa Bay's Martin St. Louis with 4:55 remaining in the third.
"It's a great feeling in that moment," Vokoun said. "It's a big momentum boost."
St. Louis also failed to score on a penalty shot against Ottawa's Brian Elliott last Oct. 29.
"I tried to get inside the glove and I shot right in it," St. Louis said. "I tried. I wasn't happy with the result."
Stillman tied it at 2 on an in-close goal at 8:53 of the third. Olesz then made it 3-2 from the top of the left circle with 7:39 to play.
Nathan Horton also scored for the Panthers, who won for just the second time (2-13-2) in 17 tries when trailing after two periods.
"A big, big victory for us, coming back," Stillman said.
Tampa Bay coach Rick Tocchet was not pleased with the performance by some of his players just two days after a 7-4 win over Southeast Division leading Washington.
"I wish we had a little more inspired play by some guys," Tocchet said.
After Halpern had a rebound goal at 9:40 of the first, Downie gave Tampa Bay a 2-1 lead from the low slot with 7:20 left in the period.
Horton put the Panthers up 1-0 from the low right circle at 5:58 of the first. The right winger has 20 goals and 34 points in 34 games against Tampa Bay.
Lightning goalie Mike Smith missed the game with a strained neck and is day to day. Smith, who hasn't practiced since getting hurt Tuesday night against the Washington Capitals, was replaced by Antero Niittymaki.
St. Louis had his 11-game points streak (11 assists, 17 points) end.
"I'm disappointed with our thought process," St. Louis said. "Everybody is working hard. You've got to work smart, especially in a 2-1 game."
The Lightning recalled goalie Dustin Tokarski from AHL Norfolk and assigned right winger Martins Karsums to HC MVD Balashikha of the KHL. ... Florida recalled defenseman Keaton Ellerby and left winger Steve MacIntyre from AHL Rochester. The pair replaced defenseman Bryan Allen and center Nick Tarnasky in the lineup. ... Matt Kenseth, the 2009 Daytona 500 winner, dropped the ceremonial first puck.