Matthews tied it 1-1 at 6:44 of the third period when he stole the puck in the corner from Zack Kassian and beat Koskinen with a shot from along the goal line.
"I thought he worked, did a pretty good job, but I thought all our guys were disconnected today," Maple Leafs coach Sheldon Keefe said. "Anything where we had success like Auston on his goal was very individual. He created the takeaway, he attacked the net, but we didn't do much with any kind of connection (throughout the game). That made it difficult because they defended well."
After Draisaitl scored 2:28 later to give Edmonton the lead, Josh Archibald shot into an empty net with 1:06 remaining to make it 3-1.
Yamamoto was credited with the goal that gave the Oilers a 1-0 lead at 10:42 of the first period after Maple Leafs forward Jimmy Vesey's clearing attempt in the slot went off the skate of defenseman Jake Muzzin and past the left pad of Andersen.
"You could tell our group had a lot more energy. The last game against [the Montreal Canadiens] (a 3-1 loss Monday), there was some fatigue in our game and it led to some mistakes," Tippett said. "Tonight, there was an excitement playing on the road, it was our first road game since last March. Pretty solid all the way through the game."
Maple Leafs forward Joe Thornton left the game at 2:51 of the third period, appearing to favor his left wrist after a collision with Archibald.
"Looks like he's definitely going to miss some time here with us, but we don't know the extent of it," Keefe said. "He's going to have to get more tests and images and stuff like that later this week. That's how it goes sometimes."
The Maple Leafs and Oilers play again here Friday (7 p.m. ET; ESPN+, TSN4, SNW, NHL.TV).