Giroux scored from above the circles at 12:34 of the first period as Travis Konecny screened Kuemper. Gostisbehere made it 2-0 on a power play at 19:10, a shot that deflected off Coyotes forward Derek Stepan's stick.
"Kind of all the bounces seemed to go their way," Kuemper said.
Galchenyuk scored a power-play goal to cut the Flyers' lead to 2-1 at 1:14 of the second period, but the Coyotes couldn't convert on two other power-play chances in the period.
"They had a lot of chances but we blocked a lot of shots and tied up a lot of sticks and we really earned that one," Pickard said. "We were good at protecting the lead, and it was a good road win."
Couturier made it 3-1 at 17:05 after Grabner accidentally upended Kuemper near the side of the net.
"I wasn't expecting an open net, and once I saw it was open I shot it as hard as I could," Couturier said.
Giroux pushed the Flyers lead to 4-1 at 6:35 of the third with his sixth goal of the season. Giroux has seven goals in 15 games against Arizona.
"We've had our ups and downs but we were able to push hard this road trip," Flyers coach Dave Hakstol said. "Obviously it's an important time of the year for us, so now we go home and have the opportunity to keep pushing."
Grabner scored his fourth of the season to make it 4-2 at 14:04, but Oskar Lindholm scored into an empty net at 18:11 for a 5-2 Philadelphia lead.
"We had our chances; they scored on theirs," Coyotes coach Rick Tocchet said.
Arizona was trying for its first six-game winning streak since Feb. 16-28, 2012. The Coyotes streak of four consecutive games with at least one shorthanded goal ended.
"We just didn't get the result we wanted, we've got to keep things rolling, keep playing the same way and have the same attitude on the road," Kuemper said.
