Steve Mason made several difficult saves late to preserve Philadelphia's fourth win in five games out West.
Mason stopped 26 shots and Philadelphia killed a late power play despite being undermanned.
But the Flyers had juice late. Simmonds tied it with 15:38 left when he knocked in a rebound on the power play for his team-high 14th goal and his ninth in as many games.
The Coyotes struggled defensively as they played without one of their top defenceman, Oliver Ekman-Larsson, who was injured Thursday on a hit by Columbus' Derek MacKenzie that earned him a three-game suspension.
Not even Doan's return from Rocky Mountain Fever could spark Phoenix.
Doan last played Dec. 4 before he started getting headaches, joint pain and severe fatigue. He was eventually diagnosed with the rare disorder that affects the lining of blood cells and is often contracted from bites by ticks.
The 37-year-old Doan never found a bite mark, but said Friday he found four ticks on his dog. After medication and rest, he began skating with the team Monday and was cleared by doctors Friday.
But Doan looked a step slow at times and his shifts were short. He logged 18:38 of ice time and his line failed to muster many scoring chances.
After a scoreless first period in which the Coyotes went 0 for 3 on the power play, the teams combined for three goals in a 1:42 stretch early in the second.
Klinkhammer and Korpikoski scored just over a minute apart to make it 2-0 Phoenix amid shaky back line play by the Flyers.
Berube immediately called a timeout, and the Flyers responded 17 seconds later when Hartnell beat Smith on the glove side to extend his points streak to seven games.
But the Coyotes answered again at 6:47. Catching the Flyers on a line change, Yandle threaded a pass to Boedker at the blue line, and he skated in alone on a breakaway for his career-high 12th goal.
NOTES: Paul Bissonnette engaged in a lengthy fight with the Flyers' Jay Rosehill in the second period. ... Hartnell has 28 career goals against the Coyotes. ... While the Flyers were returning home after the game, they play another road game Tuesday at New Jersey. ... Coyotes coach Dave Tippett said Ekman-Larsson playing Tuesday vs. Calgary is "certainly a possibility."