Jarnkrok came to prominence during the 2008-09 season, when he helped Brynas win the Swedish junior title and tied for sixth with four playoff goals. He also tied for fourth with seven assists at the 2009 IIHF World Under-18 Championship.

Jarnkrok split the 2009-10 season between the Brynas junior and senior clubs. He finished second in the Swedish junior league in points per game (1.63) and tied for 14th among junior-aged players in the Swedish Elite League with 10 points (four goals). The Detroit Red Wings selected Jarnkrok in the second round (No. 51) of the 2010 NHL Draft.

A finalist for 2010-11 SEL Rookie of the Year honors, Jarnkrok led all first-year players in goals (11) and points (27) and was tied for second in assists (16) while playing for Brynas. His three playoff goals were the most by any junior-aged player. He also had five points (two goals) for Sweden at the 2011 World Junior Championship.

In 2011-12 he tied for third in the SEL in game-winning goals (six) and was fourth in shooting percentage (14.68) as he helped Brynas win the league championship. In the 2012 postseason he ranked second among all skaters in plus/minus rating (plus-12) and was tied for second in assists (12). Jarnkrok made his World Championship debut in 2012, getting an assist and playing in all eight of Sweden's games.

With Brynas in 2012-13, Jarnkrok tied for sixth in the league in assists (29) and eighth in points (42). He saw his first North American action, playing nine games for Grand Rapids of the American Hockey League. Jarnkrok also helped Sweden win the 2013 World Championship, picking up an assist in 10 games.

Jarnkrok began the 2013-14 season with Grand Rapids but was traded by Detroit to the Nashville Predators on March 5, 2014, with Patrick Eaves and a second-round pick in the 2014 NHL Draft for David Legwand. Jarnkrok got an assist in his NHL debut with Nashville on March 21, 2014, against the Calgary Flames. He had points in his first five NHL games, scoring his first League goal March 27, 2014, against the Buffalo Sabres. Jarnkrok helped Sweden win a bronze medal at the 2014 IIHFWorld Championship.

Jarnkrok played 74 games in 2014-15, his first full NHL season. He ranked third among rookie forwards in average shorthanded ice time (1:30) and scored his first game-winning goal in the NHL on Jan. 10, 2015, against Minnesota. Jarnkrok got an assist in his Stanley Cup Playoff debut April 15, 2015, against the Chicago Blackhawks in Game 1 of the first round. He scored his first Stanley Cup Playoff goal on May 7, 2017, against the St. Louis Blues and had seven points (two goals, five assists) in 21 games, helping the Predators advance to the Cup Final for the first time since entering the NHL in 1998.

NOTES & TRANSACTIONS

• Traded to Nashville by Detroit with Patrick Eaves and Detroit's 2nd round pick (later traded to San Jose -- San Jose selected Julius Bergman) in 2014 NHL Draft for David Legwand, March 5, 2014.
• Claimed by Seattle from Nashville in Expansion Draft, July 21, 2021.
• Traded to Calgary by Seattle for a 2nd-round pick in the 2022 NHL Draft, 3rd-round pick in 2023 and 7th-round pick in 2024, March 16, 2022.
• Signed as a free agent by Toronto, July 15, 2022.