Nylander made it 4-1 at 4:50 when he shot off Koskinen's shoulder from the bottom of the right circle on the power play.
Johnsson scored his second of the game at 8:09 to put Toronto ahead 5-1 on the power play, and Tavares scored 25 seconds into the third period to make it 6-1 on a 5-on-3 power play.
"I feel like we have a lot of confidence out there," Johnsson said. "We know we haven't had the greatest starts lately, but we know if we keep grinding, we'll create opportunities and it seems like once someone starts rolling, the others will follow that guy. Marleau lifted us up with an important goal, and then Marner right away and then after that there was no stopping."
Nugent-Hopkins scored his 20th goal of the season at 16:06 to make it 6-2. Maple Leafs coach Mike Babcock challenged the play for goaltender interference, but the call was upheld after a video review.
"We've been playing some pretty strong hockey," Oilers defenseman Darnell Nurse said. "It was a [bad] second period, absolutely, but we have a game tomorrow night (at Ottawa Senators) so we need to park this one and get back to the way we played in the first period. We can sit here and whine and complain but at the end of the day, 7 o'clock tomorrow we have another game and that's where our focus needs to head to."