The League-leading Lightning (49-11-4), who have 102 points in 64 games, topped a near three-year-old record they set when they won nine in a row from Feb. 18-March 5, 2016. They have points in 14 consecutive games (12-0-2), the third longest point streak in their history.
"I didn't realize it was a franchise record, but that's pretty cool with the season we've had so far," defenseman Victor Hedman said.
[WATCH: All Lightning vs. Rangers highlights]
Hedman scored with 1:35 remaining in overtime. Nikita Kucherov had the assist, his third of the game to give the NHL's scoring leader (104 points) 23 points (eight goals, 15 assists) in his past nine games.
Tyler Johnson, Dan Girardi and J.T. Miller also scored, and Andrei Vasilevskiy made 33 saves to extend his personal winning streak to eight games.
By no means are the Lightning satisfied.
"The regular season is a dress rehearsal for the playoffs so we've just got to keep working on our game and make sure we're ready in April," Hedman said.
Mika Zibanejad, Jimmy Vesey and Boo Nieves scored for the Rangers (27-26-10), who played their first game since trading center Kevin Hayes to the Winnipeg Jets and defenseman Adam McQuaid to the Columbus Blue Jackets on Monday. Alexandar Georgiev made 26 saves.
Rangers coach David Quinn said he felt his team spent the first period looking nervous and curious about how they would play without all the players they traded, including forward Mats Zuccarello, who was moved to the Dallas Stars on Saturday.
Video: TBL@NYR: Kucherov connects with Johnson for tally
New York trailed 2-0 after the first period on goals by Johnson on a one-timer off a pass from Kucherov at 7:52, and Miller on the power play at 14:57 off Kucherov's deflected shot. It was Johnson's 300th NHL point.
"Listen, they're the best team in the League, they've got the best record, they're going to look good, but we made them look especially good because we just stood still," Quinn said of the first period. "We didn't close on anyone defensively. We didn't forecheck. We played in awe, I felt like."
Zibanejad cut the Lightning lead to 2-1 by scoring on a backhanded one-timer from the left circle 56 seconds into the second period.
"Our whole mentality changed it looked like," Quinn said. "It felt like on the bench a bubble burst and we kind of thought, 'OK, we're OK.' "
Girardi ended a 21-game goal drought to make it 3-1 at 2:53 of the second when he scored from the left circle off a cross-slot saucer pass from Steven Stamkos.
Video: TBL@NYR: Vesey battles to net, notches crafty goal
Vesey scored 30 seconds later with a power move that led to a backhand-forehand deke to make it 3-2 at 3:23. Nieves made it 3-3 with a shot that hit Vasilevskiy's right pad and the left post before banking in off the goalie's right skate at 14:14.
"From the first to the second period it felt like a totally different team," Hedman said. "We just can't turn pucks over. We're not a good team when we do that. If we play like we did in the first period when we get pucks deep and have a good forecheck we're a tough team to play against."
The Lightning still found a way to come through in overtime on Hedman's goal, a glove-side wrist shot from the right hash marks set up by Kucherov's drop pass from above the left circle.
Tampa Bay is 6-3 in games decided in overtime and 12-4 in games decided after regulation this season. Their past three wins have come in games decided after regulation, including in the shootout against the Buffalo Sabres and Los Angeles Kings.
"The swing in our game tonight hasn't happened that often, but you've got to give them credit," Lightning coach Jon Cooper said. "They found a way again."
Video: EA Sports Overtime Winner: Hedman seals the victory
They said it
"It means it's hard for teams to catch us if you're winning games like that, but more importantly it's about how you win the games. Most of the year, you leave games and you're like, 'We deserved that game.' I'm not saying we didn't deserve tonight, but I think we made it a lot harder on ourselves." -- Lightning coach Jon Cooper
"We don't quit, and I don't think we have all year. Obviously we want to have a better first period, but I think the way we battled back in the second and third just goes to show the character we have in this room." -- Rangers forward Mika Zibanejad
Need to know
Kucherov passed Martin St. Louis for the second most points in a season in Lightning history and is four points away from tying Vincent Lecavalier's team record of 108 points in 2006-07. St. Louis had 102 points in 2006-07. … Rangers defenseman Tony DeAngelo had an assist to extend his NHL career-long assist and point streaks to eight games (nine assists). … Lightning forward Brayden Point had his 100th NHL assist. … Rangers forward Brendan Lemieux, acquired from the Jets in the Hayes trade, had a game-high seven hits and blocked two shots in 15:24 in his debut.
Lightning: At the Boston Bruins on Thursday (7:30 p.m. ET; NBCSN, SN1, SNE, SNO, TVAS, NHL.TV)
Rangers: Host the Montreal Canadiens on Friday (7 p.m. ET; MSG, TSN2, RDS, NHL.TV)
Video: Hedman wins it for franchise-record 10th straight win