While Garin was rewarded for his never-say-die attitude, Chardy was his own worst enemy, issuing 13 double faults including two in the decisive third set tiebreaker.
Next up for Garin is a quarter-final meeting with Swiss Henri Laaksonen, whose stout defence made the difference in his 6-4 7-5 win over American Ryan Harrison earlier in the day on the red clay courts.
Upsets were in the humid Texas air as Norway's Casper Ruud overcame 16 aces from fourth-seeded Reilly Opelka to claw back to a 4-6 6-4 6-4 win to end the day's action.
Ruud converted his lone break point opportunity in the second set and took advantage of a mistake-prone Opelka in the third set to advance over the big-serving American.
Ruud will face Marcel Granollers in the quarters after the Spaniard upset fifth-seeded American Taylor Fritz in the opening round and crushed Australian Bernard Tomic 6-1 6-2 on Monday to advance.
With Opelka eliminated only two of the 11 American who entered the tournament are left -- top-seeded Steve Johnson and Sam Querrey.
(Reporting by Rory Carroll in Los Angeles, editing by Nick Mulvenney)
