Landeskog slipped a wrist shot past Antti Niemi for a goal that held up as San Jose's Brad Winchester hit the goal post on the Sharks' final try.
The two teams meet again Thursday in San Jose.
Dan Boyle had a goal and an assist for the Sharks, and Patrick Marleau sent the game into overtime with a tip-in at 19:38 of the third period. Jamie McGinn also scored for the Sharks, who have lost three in a row.
San Jose led 2-1 after two periods, but Colorado grabbed a 3-2 lead on goals by Winnik and Duchene. The Sharks pulled their goalie at the end of the period, a move that resulted in Marleau's score-tying goal in the closing seconds.
Scarlett Johansson steps out with fiance after pregnancy reveal
Interpol investigating stolen passports on missing flight