The 41-year-old Joseph earned his first victory since April 5 when he beat Vancouver while with Calgary. He made 29 saves.
"Wins are the most important measure for goalies, I think. Certainly it's a great accomplishment," Joseph said. "I think I really read the plays well and really focused in. That's been my problem. Sometimes you're only human and you learn."
"Cujo made three or four huge saves for us, for him, as much as anything to gather some confidence after a shaky start," Leafs coach Ron Wilson said. "And we got our offense going, we had a ton of shots, a ton of scoring chances."
Colby Armstrong tied the game 3-3 for Atlanta on a power play at 13:17 of the third, but Kubina won it by sending a slap shot past Johan Hedberg. Hedberg finished with 43 saves.
Bryan Little, with his seventh goal in four games, and Jim Slater also scored for the Thrashers, who have lost three in a row.
Joseph is scheduled to start again when the Maple Leafs host the Buffalo Sabres on Thursday.
He was making his first start since a 6-3 loss at home to the Thrashers on Nov. 25.
The Maple Leafs fell behind 2-0 early, but Joseph settled down to allow his teammates to rally.
Blake gave the Maple Leafs their first lead of the night at 3-2 with 1:26 left in the second period when he tapped in the rebound of Jonas Frogren's point shot initially deflected by Dominic Moore.
They couldn't hold that lead, but Tobias Enstrom's interference penalty with 8.1 seconds left in the third period set up Kubina for the winner.
Joseph stopped Slater on a breakaway to briefly preserve the 2-2 tie.
"They had a few chances, especially on the breakaway," Kubina said. "Cujo played a great game for us."
Joseph made a couple of solid saves before that. He was seemingly energized by Stempniak's goal at 4:16 of the second that made it 2-2.
The forward, playing on the first line with Matt Stajan sidelined by an eye injury, took a pass from Alexei Ponikarovsky and sent a wrist shot past Hedberg from the faceoff circle.
The Thrashers finished the first period up 2-1.
Little opened the scoring on a power play, completing a nifty passing play by sending Todd White's centering pass from behind the goal past Joseph at 4:41.
Just over two minutes later, Slater made it 2-0 when he took a brilliant touch pass from Joey Crabb at the blue line, walked in on Joseph and had his rebound trickle just past the goal line.
Kulemin cut into the lead at 8:29 when he converted a nice behind-the-back centering pass from Mikhail Grabovski as he cut behind then goal.
Leafs winger Jeremy Williams suffered a shoulder injury falling awkwardly into the boards early in the first period and didn't return. He came in with five goals and two assists since his recall Dec. 7. ... Stajan (right eye) and tough guy Andre Deveaux were scratched for the Maple Leafs. ... Zach Bogosian (broken left leg), the third overall pick last summer, fellow defenseman Scott Lehman (upper body) and center Jason Williams sat out for the Thrashers. ... Kulemin ended an eight-game goal-less streak.
Copyright by STATS LLC and The Associated Press
Arena: Air Canada Centre Location: Toronto, Ontario
Referees: Bill McCreary, Ian Walsh Linesmen: Scott Driscoll, Brad Kovachik
Attendance: 19,260 (102.4% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Bryan Little (19)
(Power Play) Assists: Todd White, Slava Kozlov
Jim Slater (3)
Assists: Joey Crabb, Niclas Havelid
Nikolay Kulemin (7)
Assists: Mikhail Grabovski, Jonas Frogren
Lee Stempniak (6)
Assists: Alexei Ponikarovsky, Nik Antropov
Jason Blake (8)
Assists: Dominic Moore, Jonas Frogren
Colby Armstrong (10)
(Power Play) Assists: Ilya Kovalchuk, Erik Christensen
Pavel Kubina (8)
(Power Play) Assists: Tomas Kaberle, Dominic Moore