The 12th seed cruised to a one-set advantage over the Russian qualifier, quickly securing two breaks to wrap the opener up 6-1.
However, the world number 131 looked stronger in the second as he tried to level the match.
Kuznetsov seemed to be in a good position when he broke the world number 27, but Cilic responded straight away before forcing the set into a tie-break.
The Croatian started strongly, stealing a point on his opponent's serve straight away, and Kuznetsov never looked like getting back into the tie-break.
Cilic will play Spain's Tommy Robredo in the next round.