Filip Chytil had two goals, and Artemi Panarin had two assists, extending his point streak to 10 games for the Rangers (8-7-2), who were 3-0-1 in their previous four.
Rangers starter Alexandar Georgiev made 21 saves on 26 shots in 26:42 of ice time. He was pulled after the fourth Lightning goal but returned to start the third period. Henrik Lundqvist made 15 saves on 19 shots for New York in 33:18.
"We just lost to a really good hockey team that was on their A-game, and we were far from our A-game," Rangers coach David Quinn said. "We were overwhelmed from the start all the way to the finish."
Luke Schenn gave the Lightning a 1-0 lead at 2:39 of the first period.
"Luke Schenn's goal started putting the wheels in motion for us," said Tampa Bay defenseman Kevin Shattenkirk, who had a goal and an assist. "Maybe the first few shifts weren't our best, but a simple play like that that gets to the net and goes in kind of sparked us and told us like, 'Let's start playing the way we know how. Play simple hockey and good things will come from it.'"
Killorn made it 2-0 five seconds into a 5-on-3 power play at 5:41.
Kucherov made it 3-0 with a power-play goal at 6:20, and Ondrej Palat made it 4-0 on a breakaway. It was Tampa Bay's third goal in 1:01.
"When you score often and early, it kind of helps your psyche," Lightning coach Jon Cooper said. "I don't think anybody on either team thought it was going to be 4-0 that early in the game."
Chris Kreider cut it to 4-1 at 14:33.
