David Tomasek, Adam Henrique, Andrew Mangiapane, Leon Draisaitl and Ryan Nugent-Hopkins also scored for the Oilers (4-3-1), who have won two straight.
Cole Caufield and Alex Newhook each scored twice and Josh Anderson also responded for the Canadiens (6-3-0), who had their two-game winning streak snapped.
Calvin Pickard made 22 goals to lead the Oilers to the win, while Sam Montembeault made 23 saves for the Canadiens.
Canadiens: Nick Suzuki assisted on Caufield's first goal of the night, extending his point streak to eight games. He has a goal and 11 assists in nine games this season.
Oilers: Tomasek scored his first career NHL goal in Game 7. The 29-year-old Prague native previously played for Farjestads BK Karlstad in the Swedish Elite League and led that league in points with 57 points in 47 games last season.
Edmonton trailed 5–3 early in the third period. Podkolzin played the hero, cutting in a backhand and dropping to his knees to score his first goal of the season with just 1:09 left on the game clock.
Connor McDavid had three assists, tying Jari Kurri for second most 3-point games in Oilers history.
Canadiens: Play Game 3 of a four-game series in Vancouver against the Canucks on Saturday.
Oilers: Get back on the road for a short two-game road trip starting Saturday in Seattle.
