The Kings tied the game at 2 at 17:22 of the first period on Williams' seventh goal of the playoffs.
But the Blackhawks came right back 12 seconds later on forward Patrick Sharp's fourth goal of the playoffs, putting Chicago back ahead 3-2.
It became a brand new game at 10:31 of the second period when Kings center Tyler Toffoli scored his seventh goal off a give-and-go from defenseman Matt Greene, tying it at 3-all.
Sharp then scored his second goal of the game at 18:25 of the second period on the power play, putting Chicago back ahead 4-3 heading into the third period.
"I think everybody knows it's tough to win a Stanley Cup, whether you won it the previous year or not," Sharp said. "Tonight was two great teams with a lot of guys who have gone the distance and won Stanley Cups before. I think both sides had that experience and it was a pretty entertaining game."
Added teammate and defenseman Duncan Keith, "I think it's tough to repeat, but it's tough to win in the first place, right? Like I said, it's Game 7 in the Western Conference final, round three, in overtime and it's one shot, one goal."
Game 1 and 2 of the Stanley Cup Final are Wednesday and Saturday in Los Angeles before moving to New York for Game 3 on June 9 and Game 4 on June 11.
NOTES: Attendance was a season-high 22,315. ... Prior to scoring two goals Sunday, Chicago C Patrick Sharp has been a marked anomaly in the playoffs. He led the team in goal scoring in the regular season but had just three in the first 18 playoff games. Nonetheless, Sharp came into the game tied for sixth in team history for most playoff goals (35). He is now in fourth with 37. ... Chicago is now 9-3 in its last 12 games when facing elimination, dating back to 2011. ... The Kings are now 7-11 all-time in the playoffs against Chicago, and 8-4 all-time in Game 7 playoff matches. They are the first team in NHL history to win three Game 7s on the road in one playoff year to advance to the Stanley Cup finals.