Kopitar scored both his goals during a four-goal first-period onslaught in which the Kings peppered the St. Louis net with 16 shots.
Mike Richards and Jeff Carter also scored during the span, and Justin Williams added a second-period goal. Dustin Brown racked up three assists for the Kings, who swept both games in St. Louis and will return home to host Thursday's Game 3.
Jonathan Quick earned the victory, turning aside 27-of-29 shots.
Andy McDonald notched his fifth post-season goal and Matt D'Agostini also scored for St. Louis.
Brian Elliott made 16 saves for the Blues.