Clarkson scored his first goal as a Maple Leaf and Toronto beat the New York Islanders 5-2 for its second successive victory.
The 29-year-old signed with the Maple Leafs as an unrestricted free agent in the offseason and was unable to score in his first 11 games with the team.
"The biggest thing is that we won the game tonight," Clarkson said. "I've tried to preach that the whole time. The chances were there, but I'm here to win.
"It does feel good, I guess, to get some people off my back a little bit," he joked.
Kessel also broke out of a mini-slump, scoring a pair of goals. They were Kessel's first goals in five games. He had only one point in his past seven.
"He's a highly talented player and doesn't need many chances to score goals," linemate James van Riemsdyk said. "Two sweet chances and in the back of the net.
"Sometimes it comes in streaks -- it's a long year. He's got 12 goals in 21 games. That's pretty good."
Trevor Smith had his first multi-point effort of his NHL career, scoring the first goal of the game for the Maple Leafs only 22 seconds in and adding a pair of assists for a three-point game.
He has five points in three games since being recalled from the Toronto Marlies of the American Hockey League and six points in 12 games overall this season.
"I'm just trying to fill some spots for now," Smith said. "It's always tough when you have injuries, but injuries happen. It's part of the game.
"You just got to come in, try to do the best you can, play confidently and work with what you got. The guys have been really great welcoming me and making it easy for me to play my game."
And Joffrey Lupul had three assists to end his recent slide. After starting the season with 10 points in his first nine games, he had only two points in his next nine before the assists Tuesday.
After Smith opened the scoring, the game remained 1-0 until midway through the second period.
With Cal Clutterbuck off for tripping, Kessel took a pass from Lupul, curled inside the Islanders blue line and snapped a shot from inside the right circle past Poulin's blocker hand.
It was Kessel's team-leading 11th goal of the season.
Toronto kept New York off the board until there were 12 seconds left in the second period. Cizikas got a breakaway behind the defense, deked to his left and slid the puck through Bernier's legs to cut the Leafs' lead to 2-1.
Toronto regained its two-goal lead 3:39 into the third period.
A mixup at the New York blue line by the Islanders defense gave the Leafs short a 2-on-0 break. Nikolai Kulemin was stopped by Poulin, but Raymond was able to backhand the rebound in for his eighth goal of the season.
Clarkson put the game away at 12:52, snapping a shot top corner by Poulin's glove hand.
"That was a helluva shot," Van Riemsdyk said. "He's been contributing for us in a lot of different ways out there and it's nice to see him get rewarded with a goal. I'm sure he's pretty happy about that, too."
Nielsen cut the lead to 4-2 at 15:14, but Kessel snapped home his second of the game -- 12th of the season -- less than two minutes later on a 2-on-1 break with Van Riemsdyk to make it 5-2.
Nazem Kadri sat out the third and final game of his three-game suspension.