Zach Hyman, Evander Kane and Ryan Nugent-Hopkins all contributed a goal and an assist while Jack Campbell made 32 saves for Edmonton (3-3-0).
"We were just not skating, and it seems very obvious with our group that the second we start skating and being engaged that [we] are extremely hard to stop", Draisaitl said. "It just takes us too long to get to that point as of now. That is something we can obviously work on."
Sidney Crosby picked up a goal, Rickard Rakell had a goal and an assists and Jason Zucker had two assists as Pittsburgh (4-1-1) lost in regulation for the first time this season. Tristan Jarry made 41 saves and had an assist.
"We got outplayed", Penguins coach Mike Sullivan said. "We had a real good start. I thought our first period was really strong [and] that was an area we knew we had to get better. I thought we had a real good first period, but we got outplayed from there."
Oilers "best period of the year" leads to victory over Penguins
A stretch pass from Evan Bouchard found Hyman on a breakaway and he scored 4:36 into the game to put the Oilers in front.
Rakell made it 1-1 while Pittsburgh was on the power play with a shot from the top of the left faceoff circle.
Crosby put the Penguins in front off a backdoor pass and two minutes into the second period, Bryan Rust banked the puck off Campbell to extend the lead to 3-1.
"I think the mood in the room [after the first] needed to be corrected a little bit", Kane said. "I thought the energy after the period was good and we carried that onto the ice. So, it was a good chat."
Tyson Barrie made it a one-goal game with Edmonton on the power play as his point shot went in off the combination of Brock McGinn and Jan Rutta.
Kane made it 3-3, converting a centering pass from Draisaitl and the Oilers took the lead for good when Nugent-Hopkins scored with 1:32 to go in the second.
"The second period was a good period for this home stand, and I thought we did a good job executing when we had to", Kane said. "I thought a big reason was because we got pucks to the net. It breaks down their coverage, creates some confusion in their D-zone and we got a lot of second and third chances off that, and that's something we need to continue to do."
Draisaitl evaded Brian Dumoulin and scored on the backhand with 23 seconds left.
"We just got on our heels, we didn't forecheck well and gave them some time and space", Crosby said. "They're a dangerous team and so, if you give them some time and space, they're going to make plays and we found that out."
Ryan McLeod completed the scoring with 3:15 to play, converting a drop pass from Derek Ryan.
"It's on the players. The players were the ones who decided to get it together a little bit", Oilers coach Jay Woodcroft said. "There were some things in the first period we wanted to clean up as a team, but it's the players who are the ones that go out and do that.
"Full marks to our players for playing probably our best period of the year in the second period, and that set up for success tonight."