Weise gave the Flyers a 2-1 lead at 14:45, shooting between Brossoit's pads from the top of the left face-off circle.
Oilers defenseman Darnell Nurse came close to tying the game 10 seconds into the third period, hitting the crossbar with a shot from the slot.
Raffl gave Philadelphia a 3-1 lead at 4:06 of the third on a breakaway.
"It goes both ways, a few minutes after [hitting the crossbar] they get their breakaway and then they scored," Oilers defenseman Matt Benning said. "This is a game of opportunities and whoever converts better will usually win the game. They had a power-play goal and we didn't."
Nugent-Hopkins scored on a rebound in front at 13:20 to make it 3-2.
Edmonton had a power play with 3:43 to go but could not convert. They had one shot on two power-play opportunities. Philadelphia was 1-for-4 on the power play.
"Our power play was on the ice for four minutes tonight and got one puck to the goalie's pad, that's disappointing," Oilers coach Todd McLellan said. "Their power play got nine shots in twice that amount of time. They looked dangerous and we didn't."
Forward Connor McDavid, who had the assist on Draisaitl's goal, had the only shot on the power play for Edmonton.
"He's a tough player to contain," Flyers coach Dave Hakstol said. "But I thought, overall, our players did a good job."
Simmonds scored into an empty net with 39 seconds remaining to make it 4-2.