Aston Villa striker Kodjia produced a forceful strike after shrugging his way past the home defence to open the scoring on the half hour and Crystal Palace's Zaha dribbled past three defenders before finishing with his left foot to double the lead on 70 minutes.
The Ivorians, with a caretaker coach and playing their first match since their disappointingly early elimination at the African Nations Cup finals in January, made light work of Russia, who will host the World Cup finals next year.
Russia, jeered off the field at the end of the game, have won just three games in 12 over the past year, including picking up just a single point at last year's European Championship in France.
(Reporting by Mark Gleeson in Cape Town; Editing by Neil Robinson)
