Raonic defeated Verdasco in straight sets last weekend to win the tour stop in San Jose, Calif., and the players were drawn together for the first round in Memphis.
This time they battled for three sets, with Raonic pulling out another victory 6-4, 3-6, 7-6 (7-5).
Raonic will meet Radek Stepanek in the second round.
The first round was completed with Lleyton Hewitt defeating Yen-Hsun Lu 6-4, 7-6 (7-3) and Florent Serra advancing with a walkover against ninth-seeded Xavier Malisse. Serra had a 6-2, 1-0 lead when Malisse retired.
In second-round matches, Mardy Fish outlasted Lukas Lacko 6-7 (5-7), 6-4, 7-6 (7-3) and Robert Kendrick ousted James Blake 7-6 (7-5), 6-3).
Celebrity Families of 2014 [PHOTOS]