"I feel good," said Duke, who threw five innings, allowing one run on five hits for Class A Visalia on Wednesday. "I'm progressing along, and last night I felt pretty good."
The 28-year-old struck out four and walked none.
D-backs manager Kirk Gibson said Duke is scheduled to start for Triple-A Reno on Monday at Round Rock, where he will throw 90 pitches.
"If everything goes fine," Gibson said, "He'll start next Saturday [at Houston]."
Duke, who went 45-70 with a 4.54 ERA in six seasons with the Pirates, broke his pitching hand in Spring Training after getting struck by a comebacker. He admitted the injury was a frustrating way to start his tenure with a new team.
"For sure," he said. "It's not like I blew out physically, I got hit by a line drive. That's the most frustrating part of watching these guys go out there every day, not being able to contribute or help out.
"If everything goes according to plan, I should be ready to come on back up and give this team six and seven strong."