Although he was born in Guelph, Ontario, Couture spent most of his youth growing up in Birr, near London, Ontario. He attended Prince Andrew Public School and then went on to attend A.B.Lucas Secondary School.Growing up, Logan was a Buffalo Sabres and a Dominik Hašek fan. Logan was an exceptional athlete excelling specifically in hockey, lacrosse and baseball, where he was a two-time Honda Canada/Toronto Blue Jays "hit-run-throw" contest Canadian national champion in his age group. Logan decided at age 13 to drop the pursuit of baseball for the game of hockey. He started playing minor hockey with the Lucan Irish (DD) of the OMHA's Southwestern Ontario League.He then moved to London and played three years with the 'AAA' London Junior Knights of the Minor Hockey Alliance League, where he was a teammate of current Los Angeles Kings defenceman Drew Doughty. After his Bantam year with the Jr. Knights, Couture signed with the Junior B St. Thomas Stars

In May 2005, Couture was drafted by the Ottawa 67's 12th overall in the 2005 OHL Priority Draft. Couture went on to score at a near point-per-game pace as a rookie with 64 points in 65 games in 2005–06. Couture was also selected to play for the Eastern Conference in the 2007 OHL All-Star Game, winning the shooting accuracy competition.  After completing his second OHL season, where he improved to 78 points in 54 games, Couture was drafted ninth overall by the San Jose Sharks in the 2007 NHL Entry Draft. Next season, in 2008–09, he finished in ninth place in league scoring with 39 goals and 48 assists for 87 points in 62 games. He also appeared in the 2009 OHL All-Star Game.

On October 25, 2009, Couture made his NHL debut for the San Jose Sharks against the Philadelphia Flyers. He scored his first NHL goal on November 5, 2009, against Chris Osgood of the Detroit Red Wings.Couture played his first full season with the Sharks in 2010–11 and finished with 32 goals (second among rookies) and 56 points (also second). He earned a nomination for the Calder Memorial Trophy for these achievements.