Draisaitl scored his second goal at 4:13 of the third period to give Edmonton a 3-2 lead. He has 18 points (seven goals, 11 assists) during his streak.
"That team over there, they know how to win hockey games," Draisaitl said. "For them to tie it up, and for us to come back and win the game, it's a big confidence boost for us."
WATCH: [All Blues vs. Oilers highlights]
Mikko Koskinen made 30 saves for Edmonton (27-18-6), which has won seven of its past 10 games. Ryan Nugent-Hopkins had two assists to extend his point streak to seven games (four goals, seven assists).
"We gave [Draisaitl] freebies," Blues coach Craig Berube said. "We turned the puck over and he walked in on the first one, walks right down the middle of the ice and rips one, and basically a partial breakaway on the second one."
Jake Allen made saves 31 for St. Louis (31-13-8), which has lost three of its past four games.
"We didn't really execute in that first period," Allen said. "Obviously, they came out strong and hemmed us in for most of those 20 minutes. I thought we responded pretty well in the second, but we definitely didn't have the start we wanted from myself on out, and the next thing you know, you're behind the eight ball."
Draisaitl scored at 5:31 of the first period on a slap shot from the left face-off circle to give Edmonton a 1-0 lead. Caleb Jones scored his first goal of the season 17 seconds later on a shot from the point for a 2-0 lead.
"When Leon gets going, that's a big-man's game and he played well in that game," Oilers coach Dave Tippett said. "Both his goals are good goals, they're well-earned goals. That line (Draisaitl, Nugent-Hopkins, Kailer Yamamoto) has played well for us, and it was good to see Leon has his game turned back the other way and he's playing very well."
