Allen made 14 of his 33 saves in the first period. Atkinson had a goal disallowed at 3:25 because he kicked the puck.
Allen has started nine straight games, including back-to-backs on consecutive weekends.
"I knew the week ahead I was going to play both games," he said. "Honestly, you have to prepare during the week because it's an extra start. A little more sleep, a little more rest."
Allen was on his game early, making a glove save on Markus Nutivaara on a backdoor play at 5:14.
"I knew where he was shooting. I just put my glove there," Allen said.
He then made the save against Artemi Panarin on a breakaway at 10:23 of the first.
"He's one of the best players in the League," Allen said. "He has a lot of moves in his bags of tricks. I tried to be patient. He's beaten me on the backhand before. I sort of stayed with it and stuck out my pad."
Tarasenko scored from the high slot at 9:37 of the second period to make it 2-0. It was his second goal in as many games since returning from an upper-body injury that caused him to miss two games.
Bobrovsky kept it 2-0 with a save against Tarasenko on a 2-on-1 at 14:09 of the second period, and Atkinson made it 2-1 at 14:32 when his shot from above the circles on the rush got past Allen.