The game saw the Oilers jump out to a 2-0 first-period lead only for the veteran Sharks to come back and tie the game with just over 15 minutes to play.
Milan Lucic and Oscar Klefbom scored for the Oilers before 18,347 fans at sold-out Rogers Place.
Joel Ward and Paul Martin replied for San Jose in regulation time.
It was the first playoff game in the Oilers’ new downtown arena and the team’s first NHL post-season game in almost 11 years.
The Sharks played without star centre Joe Thornton, who is day-to-day with a left knee injury.
However, San Jose centre Logan Couture returned to the lineup, albeit in a full-face cage. He missed seven games after having his mouth and teeth smashed by a deflected puck.
Watch below: Head coach Todd McLellan said the Edmonton Oilers spent 10 to 11 minutes killing penalties in Game 1 versus the San Jose Sharks
Klefbom scored the Oilers first playoff goal since 2006 when he took a cross-ice pass in the high slot from Jordan Eberle and fired the puck low between the pads of Sharks goalie Martin Jones at 6:44.
Lucic made it 2-0 with less than three minutes to go in the period on the power play.
McDavid passed the puck to Mark Letestu, who rifled the puck off Jones’s goal pad right to Lucic. Lucic then shot again through Jones’s pads to light the lamp.
The Sharks rebounded, dominating play for long stretches in the second and third periods. The teams each had 10 shots in the first period, but San Jose outshot Edmonton 28-7 through the next 40 minutes heading into overtime.
San Jose cut the lead in half 1:43 into the second period on the power play. Winger Joonas Donskoi came around behind the net and fed the puck in front on the backhand to Ward, who then redirected the puck past Oilers goalie Cam Talbot.
San Jose tied the game just over five minutes into the third period when Martin jumped on a rebound off a Tomas Hertl wraparound shot and put the puck in.
The Sharks power play, ranked an anaemic 25th in the regular season at 16.7 per cent, was otherwise ineffective. San Jose was one for six with the man advantage while Edmonton was one for three.
Game 2 goes Friday in Edmonton.