The Australian was at 15-under 127 for a one-stroke lead over Aaron Wise, the 21-year-old rookie who shot a bogey-free 63 at the new links-style Trinity Forest course.
Hometown star Jordan Spieth matched Leishman's 66 to remain eight shots back.
Leishman opened with a 61 in Trinity Forest's debut after the Nelson spent 35 years at the TPC Four Seasons. The par-70 course in suburban Irving is where Woods twice shot the previous record of 12-under 128, along with four others. Wise matched the 128 and was 14 under.
Brian Gay tied the best round of his tour career with a 62 and was two shots behind Leishman at 13 under.