Pavel Dorofeyev scored and Dylan Garand made 21 saves, but the Rangers (7-7-1) lost their fourth straight game.
New York has allowed the first goal in each of its four straight losses, including going down by multiple goals in the first period of the past three.
"The first goal is a tough one," Rangers coach Mike Sullivan said. "McDavid is a unique player. He ramps up speed like that, it's hard to stop. But other than that, I thought for the most part, I don't feel like the start was the reason. The start was scoreboard-wise, but I don't necessarily agree that it was a lack of being ready when the puck drops. I don't feel like that was the case tonight."
McDavid gave the Oilers a 1-0 lead 50 seconds into the game when he took a short pass from Bouchard in the defensive zone, banked it off the wall to himself in the neutral zone, blew right past Rangers defenseman Vladislav Gavrikov in the offensive zone and scored on Garand.
"That's your job as a leader to get us started on time," Oilers coach Mike Babcock said. "He started on time and obviously set us up for success."
Dach extended the lead to 2-0 at 6:19. Bouchard slipped a high to low pass to Dach in between the hash marks, and he then moved into the left face-off circle and scored with a low wrist shot under Garand's right pad.
"I thought that second goal was awful," Garand said. "Disappointing. It's a tough start to the game. You can't spot those guys two goals. Disappointing. I feel like I let the team down a little bit with my start tonight."
Bouchard had a would-be goal taken away for goaltender interference at 8:53.
He then scored a power-play goal at 9:49 of the second period, blasting a shot from the point past Garand to give the Oilers a 3-0 lead.
McDavid had the secondary assist on Bouchard's goal. He is the eighth different player in NHL history with an assist streak of at least 15 games at any point during a season.
Dorofeyev scored at 13:22 to cut it to 3-1. He collected the rebound from Scott Morrow's shot, slid the puck under Oilers defenseman Connor Murphy in the slot and scored on his forehand.