Mike Smith made 20 saves for the Coyotes, who ended a two-game losing streak. Phoenix won the season series from Dallas 2-1-1 and has a point in nine of 10 games against the Stars.
Jaromir Jagr scored a goal during a 5-on-3 power play for the Stars, who had a two-game winning streak snapped.
Phoenix took a 1-0 lead on a play in which four of the six Coyotes on the ice touched the puck within a handful of seconds and culminated with Doan's goal.
Steve Sullivan carried the puck into the Dallas zone, stopped in the left circle, and sent a pass back to the streaking Oliver Ekman-Larsson. Ekman-Larsson spun right to avoid the Stars defence and passed to Antione Vermette behind the net, who then fed the puck back to Doan at the right edge of the crease for the goal.
Solid passing also led to Klinkhammer's goal in his 16th NHL game — all but one with Ottawa.
Dave Moss forced a turnover behind the Dallas net, and Ben Gordon intercepted a pass near the bottom of the right circle. With Stars goalie Kari Lehtonen shading right, Gordon slid the puck to an open Klinkhammer for the goal into the vacated left side of the net to make it 2-0 at 7:31 of the second.
Jagr pulled Dallas within 2-1 at 18:19, beating Smith with a blast from the top of the right circle with 17 seconds left in Dallas' first of two 5-on-3 power plays.
Dallas had an 89-second 5-on-3 opportunity midway through the third period but were unable to solve Smith, who had allowed 16 goals in his past three starts.
NOTES: Doan has 31 goals and 37 assists in 97 games against the Stars. . Coyotes C Matthew Lombardi, who aggravated an upper body injury in Phoenix's loss to St. Louis on Thursday, was out of the lineup. LW Radim Vrbata also was out for the 10th straight game because of a lower body injury. . Stars LW Brenden Morrow was scratched due to a groin injury sustained late in a win against Los Angeles on Thursday.
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