Sanheim had tied it 3-3 at 16:00 of the third period when he one-timed a Brink pass from above the circles.
"Just trying to make a play to the net, I guess incidental contact," Sanheim said of the sequence leading to the overturned goal. "Felt like he kind of pushed his arms out too. It was kind of, I have to make a play as well."
Logan Stankoven, Taylor Hall and Jordan Staal scored in the second period for Carolina (2-0-0). Andersen made 20 saves.
Hurricanes defenseman Jaccob Slavin left the game with 3:26 remaining in the third period because of an undisclosed injury.
"He's getting looked at," Carolina coach Rod Brind'Amour said. "We'll know more about his status Monday. He came up a little gimpy. Hopefully it's nothing."
Brink had a goal and an assist, and Owen Tippett also scored for Philadelphia (0-1-1), which was 4-for-4 on the penalty kill. Samuel Ersson made 35 saves.
"I thought there was really some great efforts from some guys," Flyers coach Rick Tocchet said. "Bobby Brink was outstanding tonight, Ersson was great. The penalty kill hung in there. We’ve just got to stop giving freebie goals. That's the one thing. Can't give free goals. But other than that, I thought there was some outstanding efforts, we got 'Tipp' skating. There's some guys out there that really worked hard."
Tippett gave the Flyers a 1-0 lead with a power-play goal at 19:38 of the first period. Travis Konecny's shot from the left side bounced off a stanchion behind the net and right to Tippett, who was alone in front of Andersen.