Igor Shesterkin made 24 saves for New York, which is the No. 1 seed from the Metropolitan Division.
Game 2 of the best-of-7 series will be here on Friday.
"They play a pretty smart, simple game," Rangers captain Jacob Trouba said. "[They] don't turn many pucks over and do a good job of gaining the zone, getting pucks deep and getting in on the forecheck. I think we got to do the same. I think that's the kind of game that's going to be played. ... I'm sure we'll go back and look at it and make some adjustments for next game. ... It is one game. Come back and bring a better game in Game 2 and even this thing up."
Tkachuk gave the Panthers a 1-0 lead at 16:26 of the first period, scoring from just above the left circle after taking a pass from Gustav Forsling on a rush.
"I don't put any pressure on myself to score and produce offense," Tkachuk said. "I guess that's part of my game, a big part of my game, but at the end of the day, it's not about myself here, so there's a way bigger goal ahead."
Tkachuk leads the Panthers with 16 points (five goals, 11 assists) in 12 games this postseason.
"That was a big goal and I mean, it definitely got us going," Verhaeghe said. "Took a lot of momentum out of their sails. I think any time you're on the road and you can get the first goal, I think you take the crowd out of it a little bit and kind of ... we like playing with the lead and it kind of lets us settle down into our game. 'Chucky' was awesome."