Kucherov made it 3-1 at 7:36 of the second period with a one-timer from the left circle off a pass from Stamkos, who got his 1,100th NHL point with the assist.
Stamkos then extended the lead to 4-1 with a power-play goal at 12:02 off a cross-ice pass from Kucherov.
The No. 1 pick by Tampa Bay at the 2008 NHL Draft, Stamkos is the 66th player to reach 1,100 points (535 goals, 566 assists).
"I don't take for granted any point that I get," Stamkos said. "I've been very fortunate to play with some amazing players in my career. And like I always say, it's pretty cool to do it all with one team. And it helps to do it in a win. Everyone is in a good mood. Hopefully we can keep it going."
Mitchell Chaffee scored his first NHL goal in his fourth game at 15:18 on a rebound of Maxwell Crozier’s shot to give Tampa Bay a 5-1 lead. Chaffee was recalled from Syracuse of the American Hockey League on Wednesday.
"Four Syracuse kids on the ice for that goal (Chaffee, Crozier, Waltteri Merela and Emil Lilleberg)," Cooper said. "You talk about some of the winning we've done lately, I think you can attribute some of that to the energy that the kids from Syracuse have been bringing us. So, when they score like that, guys are pumped."
Crozier, who was called up Jan. 13, has assists in back-to-back games. Lilleberg, who was called up on Jan. 6, had the second assist on Chaffee's goal.
'It's awesome, I walked to the rink with them today, we're very close and that just makes it that much better," Chaffee said. "Crozier kind of banked it off the goalie. … The biggest thing was trying to elevate the puck, get it upstairs and it found the back of the net."
Sean Durzi cut it to 5-2 when he scored from the slot at 16:59.
Luke Glendening extended the lead to 6-2 with a shot from the left circle at 7:32.
Lawson Crouse one-timed a feed from Logan Cooley at 11:48 for the 6-3 final.
"When we mismanage our game, we pay dearly for it," Tourigny said. "I love the effort. I love our start, but unfortunately the game went away in the second period, and from there it was tough."
NOTES: Stamkos is the sixth active player to reach 1,100 points (Sidney Crosby, Alex Ovechkin, Evgeni Malkin, Patrick Kane and Anze Kopitar). ... Point has 12 points (six goals, six assists) over the past nine games. … Arizona claimed center Adam Ruzicka off waivers from the Calgary Flames on Thursday. Ruzicka had three goals and six assists in 39 games with Calgary. ... Keller is the first Coyotes player to record three consecutive 20-goal seasons since Shane Doan had nine straight from 2000-09.