Tampa Bay also got goals from Victor Hedman and Nikita Kucherov. Andrei Vasilevskiy, coming off consecutive shutouts, stopped 29 shots.
Dale Weise and Ivan Provorov scored for the Flyers, who were coming off a 3-1 win Tuesday night at Florida. Phuladelphia is 4-1-1 in the second of back-to-back games.
Steve Mason made several quality stops when the Flyers were short-handed three times in the first 12 minutes of the third before Killorn (13:28) and Callahan (13:40) scored to give Tampa Bay a 3-2 lead.
Mason finished with 25 saves.
Kucherov made it 4-2 late in the third.