Basilashvili, ranked 67th in the world, never lost his serve and allowed the 20th-ranked Karlovic just one break-point opportunity.
Basilashvili will face Matthew Ebden, a 6-4, 4-6, 6-3 winner over 2016 Memphis Open runner-up Taylor Fritz in the late match.
In other second-round action, fourth-seeded Steve Johnson beat fellow American Tim Smyczek 3-6, 6-4, 6-2. Johnson lost the first set and fell behind 3-1 in the second set before rallying.
Johnson will face Mikhail Kukushkin, a 6-3, 6-0 winner over Darian King.
The other quarterfinal matchups have No. 2 seed John Isner facing fellow American Donald Young and Ryan Harrison taking on Damir Dzumhur.
Karlovic's loss leaves Isner and Johnson as the only seeds left in the field. Four Americans have advanced to the quarterfinals, with Young and Harrison joining Isner and Johnson.