Richardson made it 1-0 at 11:58 of the first period when he exited the penalty box and scored on a breakaway for the Coyotes, who allowed two goals or fewer for the seventh time in 10 games.
"Obviously, the goals are nice and I know I have goal-scoring in me, but our whole team revolves around defense," Richardson said. "That's how we're going to have to win games, to try to get a playoff spot."
Pearson tied it 1-1 at 14:01 with his first goal since being traded to Vancouver by the Pittsburgh Penguins on Monday.
Richardson put Arizona up 2-1 at 14:45. He completed the second hat trick of his NHL career, first since Oct. 23, 2010, to make it 3-1 at 11:49 of the second off Panik's backhand pass from the goal line.
"He played a great game," Coyotes coach Rick Tocchet. "Since I took the job here (in the 2017-18 season), [Richardson is] probably the guy I can rely on (the most) on and off the ice. Any time we would get sloppy, he would go out there and settle the team down."
Galchenyuk scored on a 5-on-3 power play at 17:25 for a 4-1 lead.