The fine round was good enough to give Garrigus a two-stroke lead over Sweden's Robert Karlsson at 11-under 199.
Karlsson's birdie on the 18th hole gave him a 2-under 68 and sole possession of second at 9-under 201.
Second-round co-leader Lee Westwood posted a 1-over 71, falling into a tie for third with Heath Slocum (66). Both were sitting at 8-under 202.
Lee Janzen (70), Shaun Micheel (70), Tim Petrovic (70) and second-round co-leader Garrett Willis (73) shared fifth at minus-six.
Garrigus, ranked 377th in the world, eagled the par-5 16th thanks to a wonderful 6-iron second shot that came to within 5 feet of the cup. He drained the short putt for what was then a four-stroke lead.
Karlsson and Westwood each birdied the 16th to cut the deficit to three, Garrigus bogeyed the 18th and Karlsson chipped in for a birdie and a two-stroke deficit.