Robredo, the 2005 Estoril runner-up, rallied to oust fellow Spaniard Alberto Martin, a qualifier, 6-7 (4-7), 7-5, 6-2.
He was down a set and 0-3 in the second before mounting a huge comeback.
Also, fifth-seeded Richard Gasquet of France beat wild card Frederico Gil of Portugal 6-1, 6-2. American Vincent Spadea advanced with a 2-6, 6-3, 6-4 victory over French qualifier Olivier Patience, and 2003 Estoril runner-up Agustin Calleri was leading Juan Martin del Potro 7-5, 1-0, when his fellow Argentine retired with a fever.
In Friday's quarterfinals, Robredo faces Calleri, Spadea plays Gasquet, third-seeded Serbian Novak Djokovic plays Spaniard Guillermo Garcia-Lopez and Paul-Henri Mathieu or France meets Argentine Juan Monaco.
Notable deaths of 2014 [PHOTOS]