Almagro twice came from a break down to force a tiebreak in the first set only to lose it 7-1. He recovered, though, to dominate the baseline battle over the final two sets and complete victory in two hours 28 minutes.
It was a good day for Spain at the claycourt tournament, as 12th seed David Ferrer beat Sweden's Robin Soderling 1-6 6-3 6-2, and Carlos Moya and Oscar Hernandez won first-round ties.
There was nothing to celebrate for Britain's Andy Murray on his 20th birthday.
Murray, the world number 10, suffered a tendon injury in his right wrist that forced him to retire from his first round match against Filippo Volandri, and may well cause him to miss the French Open.
"I always try to be optimistic but I guess it's probably not looking great (for the French Open)," said Murray, who was leading 5-1 and playing superbly when he suffered the injury while attempting a routine forehand.
Robredo's defeat was the only surprise among the three second round matches on Tuesday, as seventh seed Ivan Ljubicic of Croatia beat Frenchman Florent Serra 4-6 6-2 6-2 to join Almagro and Ferrer in the last 16.
There was one upset in the first round, with Florian Mayer beating 13th seed Mikhail Youzhny of Russia 6-4 2-6 7-6, the German taking the decisive tiebreak 7-5.
The performance of the day was Moya's classic 6-0 6-3 victory over Mardy Fish.
The 30-year-old Moya, French Open champion in 1998, served impeccably and moved his opponent around the court with absolute command to seal victory in 60 minutes.
After rattling through the first set, Moya broke in the second for a 4-3 lead and held easily to leave Fish on the brink. The American saved four match points, but could do nothing when Moya took the contest on his fifth opportunity with a backhand return that flashed across his opponent.
Top seeds Roger Federer and Nadal have byes through to the second round and will not be in singles action until Wednesday.
Federer will take on Juan Monaco after the Argentine beat Dominik Hrbaty on Monday.
Nadal is on a 77-match winning streak on clay but he expects a few difficulties against his countryman Hernandez, who looked full of promise as he kept the counter-punching Benjamin Becker at bay in a 6-3 6-7 6-3 win that lasted more than three hours.
"It will be a difficult one," Nadal told reporters after practising on Tuesday. "Every court is different and the first match is always hard."
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