Horschel, who also held the lead after the second round, battled through the gusting San Antonio wind on Saturday to reach 10 under overall — two shots ahead of Jim Furyk and Charley Hoffman.
The Florida native was second at last week's Houston Open and extended his PGA Tour-best streak of consecutive made cuts to 21 this week.
Rory McIlroy shot a 1-under 71 and is four shots back. McIlroy tied for the low round on Friday with a 67, but he was inconsistent off the tee on Saturday and had four birdies and three bogeys.
Ottawa's Brad Fritsch shot 75 and is 2 over in a tie for 53rd.