"Taylor is a player who can hit very hard. He hits very hard from the baseline and I needed to change rhythm," said Granollers, who won the tournament in 2008.
"If I hit good drop shots, I think it’s a good option against him and I think today it worked."
Granollers reeled off three consecutive games, including a break at 5-2, to claim the opening set.
World number 58 Fritz bounced back in the second, surging ahead after Granollers had tied the set at 4-4.
The deciding set was all Granollers after Fritz, the 2018 semi-finalist, took the opening game.
Granollers won five consecutive games before Fritz held at 5-2.
Earlier, Norway's Casper Ruud made his River Oaks debut by defeating Bolivian Hugo Dellien 7-6(7) 6-4. His father, Christian, won the tournament in 1996 when it was a non-ATP Tour event.
Former champion Ivo Karlovic was not as fortunate as Ruud and the 40-year-old Croat, who won in 2007, fell to American Ryan Harrison 6-3 6-4.
Australian Bernard Tomic and Chile's Christian Garin also advanced.
Tomic, firing 11 aces, outlasted American Denis Kudla 7-6(8) 7-5 and Garin came from a set down to top Uruguayan Pablo Cuevas 4-6 6-4 6-2.
Top seeded American Steve Johnson is seeking to become the first player to win three consecutive titles in the tournament since Bobby Riggs did in 1936-38.
(Reporting by Gene Cherry in Raleigh, North Carolina; editing by Amlan Chakraborty)
