Zuccarello cut it 6-5 at 5:56 of the third period on a power play from the top of the left circle, and Sam Steel tied it 6-6 at 7:32 on a rebound.
Kempe answered right back 31 seconds later at 8:03 for the 7-6 final.
"Obviously we don't want to give up that many goals, but every time when they we score, we just scored right back," Fiala said. "We killed their momentum, and that was huge tonight.
"Just a good overall team win. We never gave up. We stick with it, and that's what a good team does."
NOTES: Gustavsson had an assist on Kaprizov's goal. … Doughty became the second active defenseman to score 70 power-play goals, joining Brent Burns of the Carolina Hurricanes (80). Only Rob Blake has more in Kings history (92). … With two assists, Los Angeles center Anze Kopitar passed Dave Taylor for third on its all-time scoring list with 1,070 points (365 goals, 705 assists) in 1,213 games. Kopitar trails Marcel Dionne (1,307) and Luc Robitaille (1,154).