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. Vesalainen made his professional debut at the senior level with Frölunda in the Swedish Hockey League during the 2015–16 season. After an initial loan period to return to Finland during the 2016–17 season with HPK of the Liiga, Vesalainen made his move permanent in agreeing to a one-year deal with HPK on 28 April 2017. In his first year of eligibility, Vesalainen was selected in the first round, 24th overall, by the Winnipeg Jets at the 2017 NHL Entry Draft on 23 June 2017.