Toronto center Auston Matthews missed a second straight game with a wrist injury. He is day to day.
The Maple Leafs and Oilers will play the last of a three-game series here Wednesday.
"We're in a little rut," Edmonton coach Dave Tippett said. "Teams get into a little rut, and it seems that nothing you shoot is going in the net, not even point shots or deflections. We can usually find one or two of those in a game, but it's not going in for us right now."
Mikko Koskinen allowed three goals on 10 shots before being replaced at the start of the second period by Mike Smith, who made 13 saves for the Oilers (14-10-0).
"I think once they got up three, we were getting our looks, and whenever they get a chance, they just kind of throw it into the neutral zone and have their guys skate on to it," Edmonton forward Ryan Nugent-Hopkins said. "For us, we have to get those pucks back. If our [defensemen] are going to put pucks on net, we have to find a way to get those loose pucks, maintain more [offensive] zone time and slowly break them down from there."
Toronto scored on its first two shots on goal during the three-goal first period.
Zach Hyman made it 1-0 at 7:19. He took a cross-ice pass from Rielly and lifted a backhand cutting across the goal mouth.
Nylander made it 2-0 at 10:20. He won a race to the puck in the Edmonton zone and scored on a backhand.
"I think our goalies played outstanding in both games, which helped [the shutouts] happen," Nylander said. "We've been working on our defense, and it shows a little bit in these games, which is great."