The Kings were trailing 3-2 when Kopitar got a pass from Dustin Brown and netted the equalizer 7:14 into the final frame.
Then, only 76 seconds later, Williams tipped in a Mike Richards shot, and Los Angeles hung on to even its best-of-seven Western Conference quarterfinal series 2-2 after dropping the first two contests.
Jonathan Quick turned aside 19-of-22 shots in the victory.
T.J. Oshie scored twice, David Backes also found the net and Vladimir Sobotka added two assists for the Blues.
Brian Elliott made 25 stops for St. Louis, which will attempt to regain the lead at home in Wednesday's Game 5.
2014: The Year in Music [PHOTOS]