Connor had a goal and two assists, Dubois and Ehlers each had a goal and an assist, and Connor Hellebuyck made 28 saves for Winnipeg (27-23-10), which has won three of four.
"I really liked our level, our desperation," Jets coach Dave Lowry said. "We played with lots of urgency, we played with lots of ... we had a purpose. I really felt that even giving up the goal late, we showed a lot of resiliency and stayed with our game, and we understand the value of getting that extra point."
Vladimir Tarasenko had a goal and two assists, Schenn had a goal and an assist, and Jordan Binnington made 33 saves for St. Louis (34-17-8), which was coming off a 7-4 win at the Nashville Predators on Saturday.
"Overtimes have not been good this year for us (1-6)," Schenn said. "The guys get put on the ice to do it, myself included, and we've got to be better."
Mark Scheifele tied it 2-2 at 6:30 of the third period, knocking in a loose puck near the right post following a scramble in the crease, and Connor put the Jets in front 3-2 at 7:33 following a giveaway by Marco Scandella.
"We already had two point-blank chances before that, so I think the tide was already turning for us, Winnipeg captain Blake Wheeler said.
Schenn tied it 3-3 with 2:11 remaining in the third on a one-timer from the left hash marks with Binnington on the bench for the extra skater.
"Just an entry, pretty much everyone touched the puck on the ice, and they challenged [Tarasenko], he threw it down to [Kyrou], and 'Rouzy' made a good play to me," Schenn said. "It was nice to get a point, but we've got to find a way to get two."
David Perron gave the Blues a 1-0 lead at 16:48 of the first period with a power-play goal. He scored with a one-timer from the left circle after Tarasenko's pass from along the right-wing boards softly went through the legs of Ivan Barbashev.