Shesterkin preserved the lead with a blocker save on Norris at 7:56 of the first, when he came out beyond the left post. He also stopped Claude Giroux with his glove on a point-blank shot from the bottom of the left circle at 17:10 of the second period.
They were two of his 21 saves in the first 40 minutes.
"First period, I like the way we came out," Ottawa coach Travis Green said. "They got the quick goal, but I like that it didn't faze us. I thought we got better as the game went on. Loved our second and I thought we left it out there in the third."
Lafreniere gave the Rangers a 2-0 lead at 2:56 of the third period, scoring a power-play goal on a pass that deflected in off his foot.
Reilly Smith played the puck off the wall with a pass through his legs to Filip Chytil in the right circle. Chytil then found Lafreniere in the slot after he came around the far post.
Shesterkin kept his shutout intact with three saves in a two-second span, one on Brady Tkachuk and two on Norris, during a Senators power play midway through the third.
"Those saves on the power play, that was crazy," Lafreniere said.
Gaudette, though, finally got a puck past Shesterkin at 12:28 to cut it to 2-1. He scored with a one-timer from the lower portion of the left circle off a hard pass from Tkachuk out of the far corner and through the slot.
Shesterkin stopped the final eight shots he faced with Ottawa outshooting New York 20-5 in the third.
"I thought we played an unreal game," Senators forward Michael Amadio said. "We just ran into a hot goalie."
Shesterkin's save percentage is up to .931 with 18 goals against on 262 shots in eight games. He started the night at .923.
"He's the best goalie in the League and we're lucky to have him," Lafreniere said, "but we don't want to give him that much work."
NOTES: Ottawa was 0-for-5 on the power play. It's the third time in 10 games the Senators have been held without at least one power-play goal. They were second in the NHL on the power play (42.9 percent) entering the night. … Shesterkin has made 40 or more saves in a game 15 times in his career; the Rangers are 13-1-1 in those games. … It was Lafreniere's second multipoint game this season. He also had a goal and an assist in a 6-0 opening night win at the Pittsburgh Penguins on Oct. 9.