The American saved two set points in the opening set and one in the second.
"My first two matches were pretty clean, but this match was more about fighting your way through it," Querrey said. "There were some frustrating moments against a tough opponent, but I found a way to win those tiebreaks and that's what tennis is about sometimes."
Rudd will face the winner of the suspended match between seventh-seeded Jordan Thompson of Australia and Colombian qualifier Daniel Elahi. They were tied at 3-3 in the third set when rain pushed the finish to Saturday. Galen won the first set 6-1, and Thompson the second 6-4.