The third-seeded American overcame a barrage of 35 aces and claimed victory on his third match point after playing for two hours and 13 minutes.
Roddick returned to action after missing the French Open because of a right shoulder injury.
Juan Martin Del Porto had an injury scare as he claimed a 6-4, 6-2 win over Denis Istomin of Uzbekistan.
The 12th-seeded player from Argentina called for medical attention after slipping during the fifth game of the second set. After having his left thigh taped, he closed out the match without losing a game.