Vesalainen originally played as a youth in his native Finland, appearing up to the Junior A level with HIFK before opting to continue his development in Sweden with Frölunda HC on 27 August 2015.

With the club out of playoff contention and while leading the HPK in scoring with 19 goals and 39 points, Vesalainen was loaned for the remainder of the season to league leaders Oulun Kärpät on 21 February 2018.

On 26 August 2018, Vesalainen signed a three-year, $4.476 million entry-level contract with the Jets,[5] which included an out clause that would permit him to play in Europe during the 2018–19 season if he did not make the team's roster during the training camp.

[8][9] On 21 October 2020, Vesalainen returned to HPK on loan from the Winnipeg Jets, until the commencement of the delayed North American 2020–21 season.

As an impending restricted free agent with the Jets, Vesalainen chose to return to Europe by signing a one-year contract with Swedish club, Malmö Redhawks of the SHL, on 23 May 2022.