David Tomasek, Adam Henrique, Andrew Mangiapane, Leon Draisaitl and Ryan Nugent-Hopkins also scored for the Oilers (4-3-1), who have won two in a row.
Cole Caufield and Alex Newhook each had two goals and Josh Anderson also replied for the Canadiens (6-3-0), who had a two-game winning streak halted.
Calvin Pickard made 22 stops to collect the win for the Oilers, while Sam Montembeault recorded 23 saves for the Canadiens.
Canadiens: Nick Suzuki contributed an assist on Caufield’s first goal of the night, stretching his point streak to eight games. He has a goal and 11 assists in nine games this season.
Oilers: Tomasek recorded his first career NHL goal in his seventh game. The 29-year-old native of Prague previously played for Farjestads BK Karlstad in the Swedish Elite League and led that league in points with 57 in 47 games last season.
Edmonton fell behind 5-3 early in the third period. Podkolzin played hero, sending a backhanded shot in while falling to his knees for his first goal of the season with just 1:09 remaining on the game clock.
Connor McDavid picked up three assists, tying Jari Kurri for the second-most three-point games in Oilers history.
Canadiens: Play the third game of a four-game road trip in Vancouver against the Canucks on Saturday.
Oilers: Head back on the road for a quick two-game trip, starting Saturday in Seattle.
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