Adam Lowry made it 3-1 at 10:45 on a shorthanded breakaway, and Connor got Winnipeg within 3-2 at 14:38 on the power play.
"It's nice to contribute," Adam Lowry said. "I wanted to get back that one we gave up in the first. … You're down by three, you want to obviously kill it, but if you can get one going the other way, it's huge. I thought we springboarded off of the start of the third and that helped. Huge power-play goal to give us a chance at the end."
Darnell Nurse scored shorthanded into an empty net at 19:57 for the 4-2 final.
"Everyone was calm," Woodcroft said. "We're saying the right things on the bench and we found a way to come out of tough building with two points, which is the most important thing."
Zach Hyman gave the Oilers a 1-0 lead at 12:17 of the first period, deflecting Tyson Barrie's point shot on the power play. Hyman has scored in three straight games and four of five since Woodcroft replaced Dave Tippett, who was fired Feb. 10.
"I really like how the team's playing," Koskinen said. "There's still a lot of room to improve and we try to get better every day. That's the key to be successful at the end of the year."