Roddick had not competed since suffering a hamstring pull during a second-round match against Lleyton Hewitt in Melbourne.
As the No. 2 seed in this event, Roddick received a bye past the first round and then defeated Kudla 6-7 (5-7), 7-6 (7-5), 6-4. Kudla was trying to reach the quarterfinals of an ATP Tour event for just the second time.
In first-round matches Wednesday Ryan Henderson defeated Olivier Rochus 4-6, 6-2, 6-3, Matthew Ebden won over Dudi Sela 6-7 (7-9), 6-2, 7-6 (7-1), Julien Benneteau survived against Ryan Sweeting 7-6 (7-3), 7-6 (7-5) and Denis Istomin was a 5-7, 6-3, 7-5 winner over Sam Querrey.
Roddick will meet either Istomin or Michael Russell in the quarterfinals.