Kucherov has 24 points (five goals, 19 assists) during his streak, including eight multipoint games.
Brayden Point had a goal and two assists, Steven Stamkos had a goal and an assist, and Louis Domingue made 26 saves for Tampa Bay (20-7-1), which is 8-2-0 in its past 10 games and has scored at least four goals in six of its past seven.
"People don't realize how tough it is to produce in this League, especially when you're a top guy like that and have guys on the other team trying to stop you," Stamkos said. "It's difficult and sometimes the confidence isn't always there, but when you have it, you try to keep it as long as possible.
"You can tell Kucherov] wants the puck, he's demanding it, and he's making plays. It's the most confident I've seen him this season."
***[WATCH: [All Lightning vs. Devils highlights | NHL Tonight crew talks benching of Taylor Hall]*
Egor Yakovlev scored his first NHL goal for New Jersey (9-12-5), which lost its sixth straight game (0-3-3) and was outscored by the Lightning 18-6 in three games this season (0-3-0).
Keith Kinkaid allowed four goals on 20 shots before being replaced at the start of the third period by Cory Schneider, who made five saves.
Tyler Johnson gave the Lightning a 3-0 lead at 8:20 of the second period on a tip in the slot, but Yakovlev scored off a cross-ice pass in the left circle at 11:26 to cut it to 3-1.
"The [Johnson-Point-Kucherov] line is playing well," Lightning coach Jon Cooper said. "They're just always around it and doing all the right things. And then top it off with the skill and compete and all that, as a group they're playing well and being rewarded for it."
Yanni Gourde redirected a pass from Stamkos five-hole on Kinkaid to make it 4-1 at 12:39.
Stamkos scored a power-play goal 20 seconds into the third period to extend the lead to 5-1.
