Phoenix defenseman Oliver Ekman-Larrson drilled a shot wide from the right point, and Doan converted the rebound for his first goal of the preseason.
The Coyotes forced overtime with a power-play goal at 18:17 of the third period. Kyle Wellwood scored on a backhand after a shot from Doan was not handled by Kings goalie Jeff Zatkoff.
The Coyotes also defeated the Kings 3-1 in Los Angeles on Friday.
Throughout the game at home, the Coyotes displayed improved team speed and broke in behind the Kings defense numerous times, but could not get the equalizer until late in regulation.
The Kings' goal benefited from a Phoenix turnover. Ekman-Larrson, from the right of his net, attempted to drive the puck up the boards but his pass was intercepted by Alexei Ponikarovsky, who skated into the puck and his one-timer beat Ilya Bryzgalov 12 seconds into the second period.
Scoreless through the first two periods, the Coyotes' best chance came during a second-period power play.
Adrian Aucoin closed in alone from the right point, but his wrist shot sailed right of Kings goalie Jonathan Bernier eight minutes into the period.
Early in the third period, Petr Prucha broke behind the Kings' defense and in alone on Zatkoff, who replaced Bernier, but the puck rolled off his stick before Prucha could shoot. A few minutes later, defenseman Keith Yandle broke loose across the left faceoff circle, but his shot went wide.
Phoenix outshot the Kings 38-25, and Bryzgalov, in his first outing of the preseason, earned the win and played the entire game.