Barkey's second goal of the game put Philadelphia ahead 4-3 at 11:54 of the third period. Noel Acciari knocked down an attempted pass from Damon Severson out of midair at the Columbus blue line and fed a pass to Barkey, who beat Jet Greaves from the slot to the blocker side.
Matvei Michkov and Trevor Zegras scored for the Flyers (6-10-6), who are 2-0-2 in their past four games. Joseph Woll made 28 saves.
Dmitri Voronkov, Matthew Knies and Adam Fantilli scored for the Blue Jackets (7-11-0), who have lost four straight. Greaves made 17 saves.
Voronkov put Columbus up 1-0 at 3:19 of the first period. Valeri Nichushkin's initial shot bounced off the end boards, and Voronkov buried it blocker side from the bottom of the left face-off circle for his first of the season.
Knies made it 2-0 at 4:19 with a wrist shot from the top of the right circle.
Zegras cut the deficit to 2-1 at 16:06 with a one-timer from the slot off a pass from the left wall by Tyler Foerster.
Michkov tied it 2-2 at 1:19 of the second period. He stole the puck from Fantilli at the Flyers blue line, raced down the ice on a breakaway and backhanded the puck over Greaves' left pad.
Barkey gave Philadelphia a 3-2 lead on the power play at 2:21 when he beat Greaves with a wrist shot glove side from the bottom edge of the left circle.
Fantilli tied it 3-3 at 5:44, scoring on a wrist shot high to the blocker side from the high slot.
Blue Jackets coach Rick Bowness missed a second straight game due to a health issue. Assistant coach Don Granato was in charge behind the bench.
