John Tavares gave Toronto a 1-0 lead at 4:25 of the first period. Nylander's initial shot deflected off his leg before hitting the end boards, and as it ricocheted back out in front, it hit the right skate of Smith and redirected into the net.
Edmonton has allowed the first goal in 21 of its past 25 games.
"There was lots of try in our game, lots of push, lots of people trying to have an impact on the game, which is good to see," Oilers coach Dave Tippett said. "We just can't get that play made that makes a difference on the positive side for us. (This is a) game where every play counts and you need that play to get you over the top. They got it tonight and we didn't."
Perlini tied it 1-1 at 9:58 with a one-timer off a pass from Colton Sceviour, and Draisaitl put the Oilers ahead 2-1 at 11:12 with another one-timer, this time near the left post off a cross-slot pass from Kailer Yamamoto.
TJ Brodie scored in his third straight game to tie it 2-2 at 19:32 of the first. Nylander, who had the primary assist on the play, has scored 12 points (five goals, seven assists) on a seven-game point streak.
"I'm just trying to create space and get to open areas," Nylander said. "I'm playing with great players that can find me those areas. Our line (Tavares and Kerfoot) has been getting some really good chemistry and good looks as of late, so we want to keep building off that.
"It's nice to score goals, but what's most important is getting the wins."
After Mikheyev gave Toronto the lead, Alex Kerfoot shot into an empty net with 15 seconds remaining for the 4-2 final.
NOTES:The Oilers also played without center Derek Ryan and defenseman Tyson Barrie, who were also placed in COVID-19 protocol Wednesday. … Zach Hyman was minus-3 in 20:12 of ice time in his first game against the Maple Leafs since signing a seven-year contract with the Oilers on July 28. Hyman played his first six NHL seasons in Toronto, scoring 185 points (86 goals, 99 assists) in 345 games. … Tavares has scored nine points (three goals, six assists) during a seven-game point streak.