Henman was joined by red hot German Nicolas Kiefer, who thrashed 10th-seeded Argentine David Nalbandian 6-1 6-3.
Seventh seed Carlos Moya also moved ahead with a 7-5 6-2 triumph over France's Fabrice Santoro.
In the women's draw, fourth Elena Dementieva overcame fellow Russian and U.S. Open champion Svetlana Kuznetsova 3-6 6-3 7-5.
Kuznetsova saved four match points in the final game by ripping a series of groundstroke winners before double faulting on her fifth to lose the contest.
Henman played a composed third set as he kept the Spaniard on the back foot by smartly mixing up his attack.
The Briton, who reached the final here last year, registered 40 winners while Robredo managed 35.
Despite the win, Henman was unimpressed with his form.
"My performances have definitely been a little too up and down, and there's some good bits," he said.
"But, we're all going to make mistakes and play a few bad points but I seem to be playing, two or three in a row.
"On your serve, you can't afford to do that."
Henman will play the winner of the match between 14th seed Guillermo Canas and American Taylor Dent.
Top seed Roger Federer will play later in the day.
Former champion Kim Clijsters will face former Wimbledon champion Conchita Martinez in the women's draw.
