Leonhardsen won the honorable Kniksen award as the best midfielder in 1991 and 1993. He was voted the Player's Player of the Year in Norway in 1994.
Leonhardsen started his career in Clausenengen before he, as a 19 year old, moved to Premier Division side Molde in 1989. Two years later he was brought to Norwegian champions Rosenborg. Playing for them, he was voted best player in Norway in 1994. Leonhardsen moved to Wimbledon in the Premier League the following season, where he made an instant impact. In 1998 he was bought by his childhood favourites, Liverpool. He then moved to Tottenham Hotspur, before joining Aston Villa on a free transfer in August 2002. He moved back to Norway in 2004 to join FC Lyn Oslo, where he became the captain. He signed a two year contract with second level club Strømsgodset IF from Drammen in December 2005.
Leonhardsen retired aged 37 after helping securing Strømsgodset IF promotion to Tippeligaen in 2006, and later securing a new contract in the top flight in 2007.
Now Leonhardsen is working as a youth coach in his former club Lyn Fotball.
Leonhardsen made 86 appearances for the Norwegian national team, scoring 19 goals. He was part of the 1994 and 1998 FIFA World Cup squads.
In Norway, the expression "a Leo run" is derived from Leonhardsen's running capacity and his smart movements. He is reckoned as somewhat of a cult hero and a symbol of the Norway national team success in the 1990s. He earned great respect during his career due to his highly professional approach to the game.