Jeff Carter and Dustin Penner also scored for the defending Stanley Cup champions. They erased an early two-goal deficit and a third-period deficit to earn their ninth consecutive home victory since March 23. Jonathan Quick made 19 saves.
T.J. Oshie scored his first two playoff goals for the Blues, who had the champs on the brink of serious trouble before Kopitar and Williams beat Brian Elliott, who stopped 25 shots.
Game 5 is Wednesday in St. Louis, with Game 6 back at Staples Center on Friday.