After each goalie allowed goals in the first two rounds of the tiebreaker, the Kings' Jonathan Quick and Miikka Kiprusoff of the Flames clamped down and shut out opposing skaters over the next four rounds to prolong the contest.
After Jay Bouwmeester missed for Calgary in the top of the seventh round, Williams put one past Kiprusoff and the Kings won in Calgary for the first time in 11 visits.
Rob Scudari, Dustin Brown and Williams scored in regulation and Quick finished with 32 saves for Los Angeles.
Alex Tanguay scored twice for the Flames and Robyn Regehr also found the net for Calgary, which saw its six-game winning streak snapped but still gained a point in the tight Western Conference standings.
Kiprusoff ended with 30 saves for the Flames.
2014: The Year in Music [PHOTOS]