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Wondolowski knocked a glancing header off Ramiro Corrales' long cross from the left side, redirecting the ball to the far post and out of the reach of Wizards goalkeeper Kevin Hartman.
The goal provided a surprising ending to a second half that had not featured much in the way of scoring chances, and pushed San Jose (5-11-5) above the .500 mark at home for the season.
Interim Kansas City coach Peter Vermes will have to wait at least another week to notch his first win. Vermes' second consecutive defeat was also the sixth straight match without a win for the Wizards (5-9-6).