Steven Stamkos had a goal and two assists, and Vladislav Namestnikov and Ondrej Palat each scored for the Lightning (5-1-1), who had their four-game winning streak end. Peter Budaj made 30 saves.
"We gave ourselves a chance to win," Tampa Bay forward Ryan Callahan said. "At the end of the day it's a hard-fought point. We ended up not burying on a couple power plays and they did. That's the difference."
Kucherov and Stamkos scored in the second period to give the Lightning a 4-3 lead. Kucherov beat Schneider at 14:02 before Stamkos gave the Lightning their second lead of the game 3:21 later on a backhand inside the left post.
Budaj twice denied Hischier, the No. 1 pick in the 2017 NHL Draft, from scoring his first NHL goal. The first was a glove save 17:55 into the second, the next a pad stop 6:48 into the third.
"I think one of the things you look at early in the season is if you're getting scoring chances; if not then that's when you should get worried," Palmieri said. "[Hischier] is getting chances and he's a phenomenal player. He'll find a way to put it in the back of the net. It might not be the prettiest one to get him started, but I think he has to keep playing the way he is."
The Devils took a 3-2 lead in the first period on goals by Stafford, Palmieri and Gibbons. Namestnikov and Palat scored for the Lightning in the first.
"We're all buying into the system we want to play," Palmieri said. "We haven't quite put in that full 60-minute effort the last couple of game; we were on our heels in the second [on Tuesday]. But good teams find a way to respond and our penalty kill stepped up huge and Stafford's goal on the power play was big."