Nadal leads his FedEx ATP Head2Head rivalry against Cilic 3-1 and has won their past three matches.
The second seed went on a four-game run to lead 4-2 in the second set, which included several timely volleys.
Nadal went 12/12 on net approaches during the match, fittingly hitting an overhead smash on match point to wrap up the contest in one hour and 57 minutes.
It appeared the match might head to a tie-break after Querrey dropped to 0/40 on his serve, but he regained his rhythm and fired a forehand winner on his first match point.
Kyrgios leads the FedEx ATP Head2Head with Querrey 1-0, prevailing last October in the opening round of the Shanghai Rolex Masters.