San Jose killed two penalties before Couture scored on the power play at 6:02 of the first period to take a 1-0 lead.
The Rangers' power-play struggles continued later in the first; they were 0-for-4.
"To give that team four power plays in the first period and still survive and come out 1-0 was an incredible accomplishment," Quinn said.
But Chytil got New York even 1-1 at 1:50 of the second by stealing the puck from Luke Kunin in the neutral zone and beating Reimer five-hole with a low wrist shot from the left face-off circle.
Panarin gave New York a 2-1 lead at 9:05. He connected off a give-and-go with Adam Fox with a short side high one-timer from low in the left circle.
Simek scored 4:05 later tie it 2-2 at 13:10. He came down from the point through the right circle, got the pass from Matt Nieto, went backhand-forehand and shot between Fox and past Shesterkin.
The Sharks took control of the game in the third period.
"Just a little too sloppy in transition, breaking pucks out, the passes weren't as crisp as they should be, but those happen," Rangers defenseman Jacob Trouba said. "You're not going to be on every night, but you've got to know how to play your game and adjust your game when you don't have it. You've got to simplify a little bit. I don't think we did that. I think we tried to force a little too much."
The Rangers won the opening face-off in overtime, but never got the puck out of their defensive zone. A turnover led to a Sharks' scoring chance and a save by Shesterkin. San Jose won the face-off to the left of Shesterkin, never gave up the puck, and scored.
"It felt like we played to win today and this is the way we're going to have to try and play," Karlsson said. "We're not going to win every game, but we can't be afraid of making mistakes or sitting on our heals and kind of pray that our goalies bail us out and we get an easy go somehow. I think this was a good step for us and it's all about doing it consistently."
NOTES: Panarin's goal extended his point streak to five games (11 points; three goals, eight assists), the longest season-opening point streak by a Rangers player since Brad Richards (five) in 2011-12. … Quinn's first win as a Sharks coach mirrored his first win as the Rangers coach. It came on Oct. 11, 2018, and it was 3-2 in overtime against San Jose. … It was also the first win as an NHL general manager for the Sharks' Mike Grier, who was hired July 5. … Meier played his 400th NHL game, all with the Sharks.