[5] In his final season with the Elgin Middlesex Chiefs, Jenner was named captain of the team[5] and competed at the 2009 OHL Cup Showcase Tournament.

[18] Over the final eight games of the season, Jenner tallied four goals and four assists for eight points to secure the Generals a spot in the 2012 OHL playoffs.

[29] Once they were eliminated, Jenner joined the Blue Jackets American Hockey League affiliate, the Springfield Falcons, for the 2012 Calder Cup playoffs.

[32] By December, Jenner had improved to 27 goals over 32 games and was selected to represent Team Canada at the 2013 World Junior Ice Hockey Championships.

[33] Upon returning from the tournament, Jenner was invited to participate in the Blue Jackets abbreviated training camp in January but was one of the final cuts prior to the season beginning.

In his first game back with the Generals following training camp, Jenner tallied two goals and two assists for four points in a 6–1 win over the Kingston Frontenacs.

[1] During the offseason, Jenner worked alongside Blue Jackets strength coach Kevin Collins, going through intense workouts geared toward preparing him for the NHL level.

[39] Once the season began, Jenner was placed on a line alongside Marian Gaborik and Brandon Dubinsky[40] who helped him score his first and second NHL goals on October 17, 2013, against the Montreal Canadiens.

[42] Jenner eventually returned to the Blue Jackets lineup on November 17, where he recorded five shot attempts and six hits during a 4–1 win over the Ottawa Senators.

[45] As the Blue Jackets approached the post-season, Jenner ranked second among league rookies in hits, third overall in game-winning goals, and fifth overall in shooting percentage.

[62] When the streak ended, Jenner and his second-line linemates Cam Atkinson and Brandon Dubinsky continued to produce for the Blue Jackets.

[68] Following an injury to Dubinsky in December, Jenner was moved from left wing to the centre position with Matt Calvert and Nick Foligno as his wingers.

[69] Despite the move, Jenner still struggled to score consistently,[70] until the Blue Jackets acquired Thomas Vanek after the NHL Trade Deadline.

On March 1, John Tortorella put Jenner on a line with Alexander Wennberg and Vanek and the trio began producing at a rapid pace.

[72] As the Blue Jackets qualified for the 2018 Stanley Cup playoffs, Jenner's line remained together until an injury to Wennberg forced them to add Foligno at centre.

[80] Although he missed a few games, Jenner's line continued to improve offensively although the Blue Jackets struggled to find consistency.

[78] After missing one game due to a fever, Jenner returned to the Blue Jackets lineup on March 12, 2019, where he scored his first career NHL hat-trick against the Boston Bruins in a 7–4 win.

[81] Jenner finished the season with 16 goals and 22 assists for 38 points through 77 games[1] as the Blue Jackets qualified for the 2019 Stanley Cup playoffs.

[82] In their matchup against the Tampa Bay Lightning, Jenner helped the Blue Jackets become the first team to sweep a Presidents' Trophy winner in the first round.

[89] After the Blue Jackets started the 2019–20 season with a 5–5–2 record, Tortorella reunited Jenner and Foligno with Anderson as the team's "identity line.

"[90] When this pairing failed to improve the Blue Jackets record, Tortorella replaced Jenner's linemates with wingers Gustav Nyquist and Oliver Bjorkstrand.

[95] In May 2020, Jenner was named the recipient of the Blue Jackets 2019-20 Community MVP Award in recognition of his efforts with the Nationwide Children's Hospital.

[101] After the Blue Jackets failed to qualify for the 2021 Stanley Cup playoffs due to a regular season 18–26–12 record, Tortorella was fired as head coach and replaced with Brad Larsen.

[104] However, after Laine suffered an oblique strain in early November, the Blue Jackets struggled to find a permanent replacement for him on their power play unit.

[109] At the time of the injury, Jenner led the Blue Jackets with 23 goals and 44 points and was tied with Bjorkstrand and Gustav Nyquist for fourth with 21 assists.

[113] Jenner missed 11 games after undergoing surgery but returned on January 15 to help the Blue Jackets snap a 10-game road losing streak.

[116] During the 2023–24 offseason, reports emerged that new head coach Mike Babcock requested to see personal photos of Blue Jackets players on their cell phones.

[117] While Jenner and Babock wrote a joint statement refuting these claims, Babcock resigned from his position on September 17, and was replaced with Pascal Vincent.

Prior to his rookie season with the Blue Jackets, Jeff Twohey, former GM of the Peterborough Petes, said: "He’s better, but even if he’s not ever going to be considered a great skater, he’ll figure out a way to succeed as a pro.

"[31] Jenner's uncle Billy Carroll is a four-time Stanley Cup winner, winning three times with the New York Islanders and once with the Edmonton Oilers.