Casey DeSmith allowed three goals on 15 shots before being pulled for the Penguins (26-17-9), who were playing the second game of a back-to-back following a 6-3 win at the Anaheim Ducks on Friday. Dustin Tokarski made 13 saves in relief.
Kempe put Los Angeles ahead 3-0 at 1:58 of the second period, scoring off the post from the left circle after Kopitar got to a rebound in front.
"Although we made good decisions for the most part, we come out on the wrong side of the score," Pittsburgh coach Mike Sullivan said. "They got that third goal early in the second, and it was tough to overcome that."
Kempe made it 4-0 at 5:18 when he beat Tokarski blocker side from the right circle off a return pass from Kopitar.
Kempe then completed his third NHL hat trick to extend the lead to 5-0 at 14:27. He scored a power-play goal near the right post after Kopitar's backdoor pass deflected to him off the skate of Jeff Petry.
Kempe is the first Kings player to score three goals in a period since Brown did it against the Wild on March 7, 2020.
"It was a special night for [Brown] and for us as well, so we're happy that we got the win for him," Kempe said.
Kempe scored his fourth goal of the game as a 5-on-3 power play expired to make it 6-0 at 15:09 of the third.
"Obviously, expectations go up after last year (scored NHL career-high 35 goals), not only for myself but as a team, and I think we've been getting better and better throughout the year," Kempe said. "We got to keep this going for the last 28 games, and then the fun things start after that, and that's where you got to be good. It's important to keep my scoring up as much as I can."