“That was a great gift,” James, who had lost all three of the previous NBA games he played on his birthday, told reporters.
“I had Andy in the (holiday) gift exchange and he paid me back. That’s what friends are for.”
Varejao said with time winding down on the shot clock, he did not have time to think.
“All I could do was shoot the ball and I’m glad it went in,” said the Brazilian, who finished with 14 points.
The shot was initially ruled a two-pointer, but after review officials ruled Varejao had his feet behind the three-point line.
The play helped complete back-to-back wins over the Hawks for the Cavaliers, who won 95-84 in Atlanta on Tuesday.
The Central Division leaders have now won six in a row and beaten Atlanta eight consecutive times.
James also had 10 rebounds and six assists as Cleveland won an 11th successive game at home.
Joe Johnson had 35 for Atlanta and Mike Bibby added 20.
“I think we should have won this game,” said Bibby, who missed a potential game-tying three-pointer with 4.9 seconds remaining. “I think (the Cavaliers) know that too.”
The Hawks were particularly upset that game officials did not reset the 24-second shot clock after Cleveland’s Mo Williams missed a shot, giving Atlanta, who led 99-98, about 12 seconds to shoot.
Josh Smith lost the ball to James while trying to hurry a shot, and Varejao scored on a putback to give Cleveland the lead at 100-99 with 1:31 to play.
“That was kind of like a momentum swing,” Smith said. “I thought that one official is supposed to check the clock and not all three of them (paying) attention to the game.”
