With the score tied 2-2 less than 2 minutes into the final frame, Mitchell scored on a wrist shot from the right side, breaking the Kings' 14-game streak of scoring two or fewer goals in regulation.
Brown doubled the lead 7 minutes into the period -- a score the Kings needed when the Coyotes' Daymond Langkow tallied only 33 seconds later.
Rob Scuderi got his first goal of the season and Brad Richardson scored as the Kings won for the third time in four games.
Jonathan Quick earned the win 26 saves.
Raffi Torres scored twice for the Coyotes.
Jason LaBarbera started in net for the injured Mike Smith and was chased in the third period after allowing four goals on 29 shots. Curtis McElhinney entered and stopped 6-of-6 shots the rest of the way.