Ryan McLeod and Leon Draisaitl scored in the third period for the Oilers (26-11-5), who have won two in a row after going 0-5-2 in the previous seven.
Mikko Koskinen made 25 saves for Edmonton, which also fell behind 2-0 against the Calgary Flames on Saturday before rallying to win 5-3.
"Not ideal, but it's nice we find ways to come back," coach Dave Tippett said. "That shows a lot of will on our team."
Martin made 47 saves in his second NHL start in five seasons because top goalies Thatcher Demko and Jaroslav Halak are in NHL COVID-19 protocol. Martin got his first NHL start Friday against the Florida Panthers, making 33 saves in a 2-1 shootout loss, but was put in protocol Saturday and couldn't play against the St. Louis Blues on Sunday before being cleared Tuesday.
"I'm pretty excited for these opportunities," Martin said. "It was a slight letdown when you test positive, I'm sure for any guy who goes through that, and then to be back in the lineup felt great. … I feel really good right now but having a 2-0 lead going into the third period, you want to close that out every time so it's frustrating to not be able to finish that off."
Martin made a good impression on McDavid.
"It seems to go either two ways with guys you don't know, they either the lay an egg or they're amazing," McDavid said. "Tonight he was amazing. We had lots of chances and he was in the right position all night and we had to work for our goals."
Elias Pettersson and Tyler Motte scored 2:47 apart in the second period for the Canucks (18-19-5), who went 0-1-1 on a three-game homestand. Vancouver has two wins in its past eight games (2-4-2) after starting 8-0-1 under coach Bruce Boudreau, who replaced Travis Green on Dec. 5.
"It hurts," Motte said. "It's one you've got to finish off. We know we've dug ourselves a hole, we've got to get points and that's a great example of a time we've got to finish the job."
Pettersson put the Canucks ahead 1-0 with a solo rush at 14:26 of the second period, scoring on a wrist shot from the top of the left face-off circle.