Thai fourth seed Paradorn Srichaphan was earlier knocked out 6-4, 6-3 by Russian Nikolay Davydenko, who next meets Jarkko Nieminen after the Finn outlasted Dutch eighth seed Martin Verkerk 7-6, 6-7, 6-4.
Henman, three times a finalist here, struggled in the first set against Robredo's baseline game and wasted a set point in the 10th game before converting two games later.
He broke Robredo in the fourth game of the second set and from there the Spaniard could not find a way back as Henman clinched the match.
"It felt good tonight," said Henman, "It has been a couple of weeks since I played a match and my serving was very consistent.
"This match gives me confidence. I feel happy about the improvements I made. I think this was a good quality match," he added.
The British world number 11 will meet the winner of Wednesday's match between Sweden's Jonas Bjorkman and Czech qualifier Tomas Cakl in the next round.
Paradorn was joined at the exit by Ivo Heuberger after the Swiss pulled out of his match against former French Open champion Albert Costa with a shoulder injury when he was trailing 6-4, 3-0.
Paradorn started poorly, dropping the first four games of his match against a Russian opponent who, at world number 51, is ranked more than 40 places below him.
"I really moved too slow today," Paradorn said. "I tried to fight back after an early double break but I missed the small chances I had on his service."
"Here... my first time in Rotterdam, where everything looks okay and we play in a fine stadium... it simply did not work out today."
Raemon Sluiter thrilled home fans with a comprehensive victory over Alex Corretja, the Dutchman beating the former world number two 6-4, 6-2 to advance.
