Bouchard gave the Oilers a 3-2 lead at 10:36 of the third period, beating Markstrom with a slap shot from the point on a power play.
Nugent-Hopkins scored an empty-net goal at 19:17 for the 4-2 final.
"I don't think we're happy with our second," Edmonton forward Derek Ryan said. "They took it to us a little bit. That's what happens in big games, a lot of momentum swings and ebbs and flows. We were able to come out and have a really good third, I was happy about that, and regroup and found two points."
Draisaitl put the Oilers ahead 1-0 at 19:44 of the first period. He scored on a one-timer from the right face-off circle off a pass from Nugent-Hopkins on a power play.
Brown extended the lead to 2-0 at 3:13 of the second period, chipping in a loose puck at the top of the crease.
"Up and down the lineup ... first line, fourth line, all the [defensemen], both goalies ... you're going to need everybody," Pickard said. "It's very exciting and encouraging going forward. We want to finish business here for the next few games. We're obviously playing for seeding and want that home ice. It's an exciting time of the year."