Williams scored his second goal with 20.4 seconds left and the Blackhawks snapped a nine-game losing streak in Dallas, beating the Stars 5-4 on Wednesday night.
Dallas scored three straight goals in the third to erase a 3-1 deficit.
Williams tied the game before netting the winner to give Chicago its first win in Dallas since 2001. He also beat the Stars with an overtime goal Oct. 13.
"This time we found a way to win," Williams said. "We gave up a couple, then took the momentum back. We're making it hard on ourselves."
No one experienced a wider range of emotions than Modano.
Modano was initially awarded an assist on Sergei Zubov's power-play goal in the third for his 1,232nd point, matching Phil Housley for the most points by an American-born NHL player. After the game, it was ruled Modano never touched the puck and the assist was taken away from him.
This probably explains why Modano was hesitant to acknowledge the crowd after he appeared to tie the mark. He said the official scorers shouldn't have credited him with an assist in the first place.
"Someone screwed up," Modano said. "It's partly why I didn't want to stand up. After a moment I stood to get it over with and get the game running again."
The Zubov goal at 10:07 gave the Stars a 4-3 lead, and it appeared the Stars were going to turn back the Blackhawks once again in Dallas.
But Williams made sure the Blackhawks left Texas with a win this time. Chicago last won in Dallas on March 7, 2001 -- back when the Stars played at Reunion Arena.