Early NHL career: Calder Trophy, contract extension, and franchise records

On 13 July 2020, Kaprizov signed a two-year, entry-level deal with the Minnesota Wild.^[10] The contract included the 2019–20 season, which was impacted by the COVID-19 pandemic, but he did not play his first game until the 2020–21 season.^[11] Kaprizov made his NHL debut on 14 January 2021 against the Los Angeles Kings. He tallied three points, including the game-winning goal in overtime, in a 4–3 victory for the Wild. In this game, he became the first player in NHL history to score three points and an overtime goal in his debut,^[12] and became the third player in NHL history to score an overtime goal in his debut.^[13] He was subsequently named NHL First Star of the Week for the week ending 17 January.^[14] Kaprizov scored his first NHL hat trick on 12 March, in a 4–0 victory over the Arizona Coyotes.^[15] On 19 April, Kaprizov scored his 37th point, breaking the Wild's franchise record for points by a rookie, which was previously held by Marián Gáborík (in 2000–01). Despite the shortened 2020–21 NHL season as a result of the COVID-19 pandemic, Kaprizov set Wild franchise records for goals and points by a rookie, with 27 goals and 51 points. He was awarded the Calder Memorial Trophy as rookie of the year, becoming the first player in Wild franchise history to receive the honor. He received 99 out of 100 first place votes, the highest percentage since Teemu Selänne in 1993.^[16]

On 21 September 2021, Kaprizov signed a five-year, $45 million contract with the Wild.^[17] The contract came after months of negotiations in which Kaprizov was reported to be willing to return to CSKA Moscow if did not come to terms with the Wild.^[18] Regarding these reports, Kaprizov said there was "no chance" he would have returned to the KHL.^[19] Upon signing the contract, Kaprizov became the highest-paid sophomore player in NHL history.^[20] After starting the season with 40 points in 32 games, Kaprizov was selected for his first NHL All-Star Game.^[21] At All-Star weekend, he participated in the Breakaway Challenge in the NHL All-Star Skills Competition with a tribute to Alexander Ovechkin, wearing Ovechkin's jersey and performing his famed "hot stick" celebration.^[22] On 2 April 2022, Kaprizov broke Marián Gáborík's franchise record for points in a season, with 84. Later that month, he went on to break Gáborík's franchise record for goals in a season, with 43.^[23] By the season's end, Kaprizov had set Wild franchise records in each of the major scoring categories with 47 goals, 61 assists, and 108 points.^[24]

During the 2022 off-season, Kaprizov returned to Russia to visit family, despite Wild general manager Bill Guerin encouraging him to stay in Minnesota due to the ongoing Russo-Ukrainian War.^[25] In July 2022, multiple Russian media outlets reported that Kaprizov was wanted in Russia for allegedly purchasing a fraudulent military identity card in 2017 to evade the mandatory military service required of Russian men aged 18 to 27.^[26] The reports occurred days after Philadelphia Flyers prospect Ivan Fedotov was arrested in Russia for purchasing a military ID in 2017.^[27] Kaprizov and Fedotov were teammates on Salavat Yulaev Ufa during this time. Responding to these reports, Kaprizov's father denied the allegations and stated that Kaprizov was a student at the Russian Presidential Academy of National Economy and Public Administration, and thus his military obligations were delayed as a result of his studies. Despite the reports being largely refuted by Western media, Kaprizov unsuccessfully attempted to go to the United States twice in the days following, first via Dubai and then via the Caribbean. He returned to Russia after these attempts. On 1 August, Kaprizov successfully entered the United States via Turkey, arriving in New York City before returning to Minnesota the following day.^[28][29]

Unlike the 2021-22 season in which he did not score until the 9th game of the season, Kaprizov began the 2022-23 season with 8 goals in the Wild’s first 10 games. While facing the Los Angeles Kings on November 9, Kaprizov received a match penalty for cross-checking Kings defenseman Drew Doughty in the face; Kaprizov was ejected as a result of the match penalty and fined $5,000 by the NHL.^[30] After beginning the season with 22 goals and 47 points in 37 games, Kaprizov was named to his second consecutive All-Star Game.^[31] Kaprizov suffered a lower-body injury after Winnipeg Jets defenseman Logan Stanley fell on him during a game on March 8. The injury forced Kaprizov to miss 13 games in March and April.^[32] Despite the injury limiting him to 67 games played, Kaprizov reached 40 goals for the second consecutive season, becoming the first player in Wild history with multiple 40 goal seasons.^[33] In a first-round playoffs against with the Dallas Stars, Kaprizov was largely held in check by the Stars defense, scoring only 1 goal and recording 0 assists as Dallas won the series in six games.^[34]

Following the 2024 NHL All-Star Game break, Kaprizov was joined on the Wild's top line by Joel Eriksson Ek and Matthew Boldy. By 15 February, Kaprizov and Eriksson Ek had added five points over three games and the trio had combined for 17 points.^[35] On 19 February 2024, Kaprizov and Eriksson Ek scored hat tricks in the third period to lift the Wild 10–7 over the Vancouver Canucks. As a result of their hat-tricks, they helped set a new team record for most goals in a game and matched the franchise record for points in a game.^[36]