The American saved two set points in the opening set and another 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."
After going 4-1 up in the first tiebreak, the 71st-ranked Querrey dropped five consecutive points to the 372nd-ranked Serb.
Querrey rebounded to win the next four points and the set.
The American let a 4-1 lead almost slip away in the second tiebreak as Tipsarevic again rallied.
But the final three points and the match went to Querrey.
The twice Houston finalist will play Chile's Christian Garin in Saturday's semi-final.
Garin breezed past Swiss qualifier Henri Laaksonen 6-3 6-2.
Norway's Casper Ruud also made the semi-finals after defeating 2008 champion Marcel Granollers of Spain 6-1 6-0 and will play either Australian seventh seed Jordan Thompson or Daniel Elahi Galan of Colombia, who had their match suspended due to rain.
Galan dominated the opening set 6-1 win before Thompson took the second 6-4. The two will resume play on Saturday with their third set even at 3-3.
(Reporting by Gene Cherry in Raleigh, North Carolina; Editing by Toby Davis/Peter Rutherford)
