Hayes skated the puck from the red line into the right side of the Blue Jackets zone before beating goalie Elvis Merzlikins with a wrist shot from just inside the left face-off circle.
Philadelphia (34-20-7) swept a home-and-home and the season series against Columbus (30-19-13), which moved into the first wild card into the Stanley Cup Playoffs from the Eastern Conference despite losing its seventh game in a row (0-3-4).
WATCH: [All Flyers vs. Blue Jackets highlights]
The Flyers have a two-point lead for third place in the Metropolitan Division on the Blue Jackets, who they defeated 5-1 in Philadelphia on Tuesday.
"It's going to be a fun run if we play the right way," Hayes said. "I thought we played the right way the last two games. Four points against a division opponent that was above us and now is beneath us.
"Two or three weeks ago, we were battling just to get in the wild card, and now we're battling for home ice."
Columbus had not lost seven straight games since starting the 2015-16 season 0-8-0.
"I thought we hung in there," Blue Jackets coach John Tortorella said. "We played hard. Found a way to lose it at the end."
Travis Konecny had a goal and two assists for the second straight game, and Brian Elliott made 28 saves for the Flyers, who are 5-2-0 in their past seven games and 7-3-0 in their past 10.