The 23-year-old goalie is 2-0-2 with a 2.00 goals-against average, .914 save percentage and one shutout this season. He could have become a restricted free agent after the season.
"He said it's a big step in his life and he's very happy about that, and he thanks Carolina for the trust," Kochetkov said through translation by Hurricanes forward Andrei Svechnikov. "He couldn't believe it at first and he still can't believe it. He said it is a big step for him, playing here, and he feels more confident and he's only going to get better and better."
Kochetkov made his NHL debut last season when goalie Frederik Andersen was out with an injury. He was 3-0-0, allowing six goals on 61 shots (2.42 GAA, .902 save percentage) and at 5-0-2 for his career, he became the 13th goalie in NHL history to help his team earn at least one point in each of his first seven games.
"He's shown in the little time that he's been here that he can play," Hurricanes coach Rod Brind'Amour said. "(It's) the whole makeup of the player, his personality, his willingness to work hard and obviously his ability. Goaltending, we know, is so important, so it's nice to have him in the mix."
Last season, Kochetkov was 13-1-1 during the regular season for Chicago of the AHL and helped them win the Calder Cup when he was 5-1 with a 1.65 GAA and .950 save percentage in six playoff games.
"Pyotr has been extremely solid for both Carolina and Chicago (of the American Hockey League) since coming to North America last season," Hurricanes general manager Don Waddell said. "We believe he has a bright future in our crease, and we're excited to watch him as he continues to grow as a person and player."
Carolina selected Kochetkov in the second round (No. 36) of the 2019 NHL Draft. He got his first NHL shutout Nov. 14, when he made 27 saves in a 3-0 win against the Chicago Blackhawks.
Andersen (5-3-0, 2.72 GAA, .891 save percentage) is currently out with an undisclosed injury. Antti Raanta is 3-2-2 with a 2.53 GAA and .902 save percentage.
Kochetkov started for the fifth time in seven games for the Hurricanes (10-5-4) when they hosted the Arizona Coyotes on Wednesday.
"It's not just about the ability," Brind'Amour said. "You've really got to get a feel that this kid's going to want to be that player. I feel pretty confident we've got a good one here."
NHL.com independent correspondent Kurt Dusterberg contributed to this report