Tyler Johnson, B.J. Crombeen and Ondrej Palat scored in regulation for the Lightning, who had lost each of their previous five home games (0-3-2) by one goal. Victor Hedman, who sat out a game with a leg injury, had three assists in his return.
Neil tied it at 3 from the right circle with 2:18 remaining in the third.
Tampa Bay went ahead 3-2 at 5:31 of the third when Martin St. Louis extended his point streak to nine games by sending a pass through the slot to Palat, who scored from just below the left circle.
St. Louis (963) moved within two points of tying Maurice Richard (965) for 85th on the NHL list.
Crombeen's redirection put the Lightning up 2-1 at 3:30 of the second. MacArthur got the Senators even at 2 by putting an in-close power-play shot past Bishop with 3:03 left in the second.
Ottawa's power play, the NHL's top-ranked road unit, had mixed results in its second opportunity in the first period. After Johnson beat Anderson for a short-handed goal at 14:32, Karlsson tied it at 1 on his third goal and 23rd man-advantage point of the season 40 seconds later.
Anderson was coming off a 2-0 victory Tuesday at Washington, which was the Senators' second straight road shutout.
NOTES: Tampa Bay star Steven Stamkos, sidelined since breaking his right shin on Nov. 11, was OK after taking a spill during a drill at the team's morning skate. "I was kind of laughing," Stamkos said. "It's the first time I've fallen in 10 weeks. Just knowing that everything was fine, you get right back up, I didn't feel anything whatsoever. So, that's nice." Stamkos, who hasn't ruled out playing in the Sochi Olympics next month, was scheduled to have X-rays Thursday night to see how the leg is progressing. ... Ottawa coach Paul MacLean used the same lineup as the Washington game, including D Marc Methot being scratched. ... Chicago Blackhawks senior adviser and 13-time Stanley Cup winner Scotty Bowman, a regular at Lightning home games, is scheduled to have right knee replacement surgery Friday in Tampa.