Kuznetsov failed a doping test after a sample was taken on May 26, the day Russia won the bronze medal game against the Czech Republic in Slovakia, the International Ice Hockey Federation said on Friday.
The previous day, Russia lost in the semifinals to eventual champion Finland.
The ban does not stop Kuznetsov playing in the NHL, which classifies cocaine as a drug of abuse rather than a performance-enhancing doping product.
"I feel absolutely terrible for letting you down," Kuznetsov said in a statement published by the NHL addressing Capitals owners and fans.
"I realize that the only way I can win you back is to take ownership of my situation and my actions from this point forward," said the 27-year-old player, who has six years left on a $62.4 million contract signed in 2017.
The NHL said Kuznetsov agreed to meet with commissioner Gary Bettman before training camp, and will undergo regular testing.
"Here, we understand that Mr. Kuznetsov has voluntarily sought help through the education and counseling program provided for in the NHL and (players' union) collective bargaining agreement," NHL deputy commissioner Bill Daly said in a statement.
Kuznetsov faced allegations of drug use at the world championship after a video posted on social media showed him talking to someone while lines of white powder and American dollar bills could be seen on the table. However, it was unclear when the clip was filmed.
Kuznetsov, who did not touch the powder in the soon-deleted video, said it was from 2018 in Las Vegas after the Capitals won the Stanley Cup with a standout performance from the Russian. He said it was a friend's hotel room and he soon left after seeing drugs there.
"I never took drugs, give me a drug test and I'll pass it," he told Russian media outlet Sport Express in May.
The NHL closed its review of the incident within days, saying it found no reason to question the player's explanation.
Russia's sports minister suggested the United States should take some blame for drug use by Kuznetsov, who made his Washington debut at age 21.
"It's not possible for us to control athletes who at a young age leave Russia for places with a completely different culture," minister Pavel Kolobkov said in a statement.
Kuznetsov's ban expires on June 12, 2023, the Zurich-based IIHF said.
The two-time world champion said he would "accept this penalty." He could have pursued an appeal to the Court of Arbitration for Sport.
He is currently excluded from the 2022 Beijing Olympics and the next three world championships.
Though the NHL did not release players for the 2018 Olympics in South Korea, the next Winter Games in China's capital city is a more attractive marketing prospect to justify shutting down league play for most of February.
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