Brandon Hagel scored, and Andrei Vasilevskiy made 21 saves for the Lightning (27-19-5), who had a three-game winning streak end in their return from the NHL All-Star break.
The Lightning hadn't played since Jan. 27.
"They had a little bit more in the tank than we did at the end," Lightning coach Jon Cooper said. "They just made a couple more plays that we couldn't make. I thought we got a little bit tired as that game went on."
Lightning defenseman Mikhail Sergachev had to be taken off the ice on a stretcher with 13:00 remaining in the second period, when his left leg bent awkwardly underneath him as he was going for a hit on Alexis Lafrenière.
Sergachev was playing in his first game since Dec. 19 after missing 17 with a lower-body injury.
"It's super tough," Lightning defenseman Victor Hedman said. "Obviously, been through it with a few other guys in the past, but [Sergachev], first game back after missing an extended period of time, to see him in such pain, it's hard. You feel for him and you just pray for the best, but obviously it's not looking good."
With Quick in net making 32 saves, New York won 2-1 in overtime against the Colorado Avalanche on Monday, its first game back from the All-Star break. The Rangers went into the break with a 7-2 win against the Ottawa Senators on Jan. 27, also with Quick in net making 29 saves.
Laviolette said Wednesday morning the plan was always to start Quick in the first two games after the break to give Igor Shesterkin more time to work with goalie coach Benoit Allaire.
He did not say who would start at the Chicago Blackhawks on Friday.
"I think we're playing better as a team in front of him, but I would say he's been really good," Rangers defenseman Jacob Trouba said of Quick. "When we need him to make a big save he's made them. Everyone in here rallies around what he's doing for us right now."
Vesey gave New York a 1-0 lead at 1:15 of the second period, when his backhand shot from low in the right face-off circle went over Vasilevskiy's left shoulder and into the net.
Quick stopped Steven Stamkos on a semi-breakaway 42 seconds earlier.
Sergachev went down less than six minutes later, and Cooper said the Lightning were "clearly affected" by what happened.
"You want to try to rally behind that, but we're humans first and when you see something like that it puts an uneasy feeling in your stomach so it puts the game into perspective," Stamkos said. "We just hope he's OK."
Brodzinski made it 2-0 at 17:46, scoring with a one-timer from the right circle off a pass from Chris Kreider, who got the puck after Blake Wheeler dug it out of a scrum by the boards.
"I found a little soft spot and I just tried to put something on net," Brodzinski said. "It squeaked through."