Green gave the Red Wings a 1-0 lead at 14:12 of the first period, beating Quick with a wrist shot for his second goal of the season. Frans Nielsen's backhand pass set up a 2-on-1 and Green waited for Kopitar to commit before shooting.
The Red Wings nearly took a 2-0 lead in the second period after a defensive mix-up by the Kings power play, but Quick made a lunging blocker save on Dylan Larkin's point-blank shot. Quick stopped Nyquist on a breakaway later in the period.
"We had enough chances to score more than one goal, but he's a good goalie who makes those crazy saves," Nyquist said.
Los Angeles took a 2-1 lead with goals later in the period.
"That was huge, because we were losing 1-0 and we had nothing going on," Kopitar said. "We gave them a great chance on our power play, but [Quick] makes that save and a couple more big ones, and we were able to come back."