The world number 74 cruised to a 6-3 6-2 triumph in the first meeting between the pair, needing a little over an hour to finish the Spaniard off at the Jockey Club Brasileiro.
Haider-Maurer broke early in the first set to seize control and also claimed a break in the second set, without allowing a single break point on his own serve throughout the match.
“It was a great win for me, for sure,” Haider-Maurer told the ATP Tour website. “I played well against Tommy Robredo, and I’m very happy… Every round here is very important. Every round is a big step.”
His prize is a quarter-final against Brazilian wildcard Joao Souza, who had to come through a war with Slovakian Blaz Rola 6-4 6-7 (9/11) 6-4.
Second seed David Ferrer was edging a stern test of his credentials before Dutch qualifier Thiemo de Bakker had to retire due to stomach issues.
Ferrer was 7-6 (10/8) 2-0 up at the time and he will now meet Argentinian Juan Monaco in the last eight after the world number 66 saw off Finland’s Jarkko Nieminen 6-1 7-6 (7/4).
In the women’s draw Romanian Irina-Camelia Begu remains in contention after battling past Bethanie Mattek-Sands 6-2 6-4.
Begu, who also reached the quarters at last year’s tournament, needed just under an hour and a half to beat the American and equal her 2014 efforts.
“It was a really tough match because Bethanie’s a great player, even if she’s now quite low in the rankings,” Begu told the WTA Tennis website. “I knew from the beginning that I’d have to fight for every ball and in the end I won – so I’m happy.”
Julia Glushko now stands between Begu and a place in the last four after she reached her first-ever WTA quarter-final by knocking out American Grace Min 6-3 6-1.
Rounding off a bad day for the Americans fourth seed Madison Brengle bowed out to Dinah Pfizenmaier 7-5 0-6 7-5.
Pfizenmaier will need to claim another scalp if she is going to progress further, with fifth seed Johanna Larsson set to face her in the last eight after overcoming a hostile crowd to beat Brazilian Gabriela Ce 7-6 (7/4) 6-4.
“It was a true battle. The crowd was so noisy and she was fighting until the end. I had to play my best to win and I’m pleased with how I played at the important moments,” the Swede told the WTA website.
Elsewhere, sixth seed Anna Schmiedlova eased through 6-3 6-0 against Alison Van Uytvanck and seventh seed Polona Hercog overcame An-Sophie Mestach 6-3 6-4.