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.
The American sprinted to a 5-3 lead in the second set, but showed his first sign of nerves in trying to serve out the match.
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.