Callahan gave the Lightning a two-goal lead at 11:58, but had to sweat out the final 56 seconds in the penalty box after being called for slashing.
Ottawa struck once with goalie Robin Lehner pulled for an extra attacker, but couldn't convert on the 6-on-4 advantage.
Teddy Purcell scored twice in the first period for the Lightning, who matched a season high with a fifth straight win. Birthday boy Valtteri Filppula supplied the other Tampa Bay goal, while Ben Bishop made 27 saves in his 100th NHL game.
Lehner surrendered five goals on 28 shots in the Senators' fifth consecutive loss.
Celebrity Families of 2014 [PHOTOS]