The 35-year-old Haas, who won the tournament in 2006, lost just one of 20 points on his first serve against Odesnik and converted five of six break-point chances.
No. 2-seeded John Isner had 20 aces and held every service game in the final two sets to beat fellow American Michael Russell, 4-6, 7-6 (3), 6-4.
Wild card Marcos Baghdatis lost only five points on his first serve and took advantage of 10 double-faults by Jiri Vesely to win 3-6, 6-2, 6-1. Qualifier Steve Johnson of the United States eliminated Mikhail Kukushkin 6-4, 6-3.