French players Jo-Wilfried Tsonga, Julien Benneteau and Michael Llodra joined Zverev in the next round. Llodra edged Marcos Baghdatis of Cyprus 7-6 (6), 6-4, and in two all-French matchups, Tsonga topped Josselin Ouanna 7-6 (9), 6-4, and Benneteau beat Florent Serra 6-4, 6-4.
Zverev converted both breakpoint chances in the third set to build a commanding 5-0 lead, while the fourth-seeded Robredo wasted three opportunities.
The German left-hander believed in his chances despite starting the season with five straight losses.
"I felt like I had a chance, no matter what," Zverev said. "I just tried to mix it up and not give him too many easy balls."
Zverev had lost both of his previous meetings with Robredo.
"I know his game," Zverev said. "I can use my strokes against him and it worked out."
Defending champion and second-seeded Tsonga did not give any break chances to Ouanna, firing 19 aces to win a tight tiebreaker and take a 3-1 lead in the second set.
"It wasn't great tennis but I got the main thing [the victory]," Tsonga said. "Getting used to the lights and to the speed of the ball was a bit tough."
Meanwhile, Llodra hit a service winner to convert his third match point. The 2009 finalist broke Baghdatis in the previous game with a backhand return winner.
"It could have gone either way," Llodra said. "I tried to be more aggressive and I seized my chances."
The seventh-seeded Baghdatis had two set points in the tiebreaker but squandered them, allowing Llodra to take the opening set with a smash.
In another second-round match, the eighth-seeded Benneteau broke Serra at 4-4 in the second set and saved two break points in the next game to hold serve and clinch victory.
Benneteau relied on a strong serve, landing 76 percent of his first serves and slamming 11 aces.
Also Wednesday, Belgian qualifier Yannick Mertens beat Jan Hajek of the Czech Republic 7-6 (8), 6-1, to set up a second-round match with Guillaume Rufin, who outlasted fellow Frenchman Laurent Recouderc 7-5, 4-6, 7-6 (2).
Lucky loser Recouderc replaced Mikhail Youzhny as the sixth-seeded Russian had not recovered from the groin injury he sustained last Sunday in the final of the ABN Amro in Rotterdam.