Nalbandian, last year's runner-up who won his 10th career title last month, appeared to be headed for a romp after winning the first set, but the Argentine's serve worsened and Koellerer was able to take advantage to win only his sixth ever match on the main tour and his first since July.
Koellerer broke early in the third set and clung to the advantage, saving break points at 3-2 and 4-3 against the Argentine for the biggest win of his journeyman career.
Almagro, the fourth seed, spoiled Santiago Gonzalez's 26th birthday with a 6-4, 6-2 win over the Mexican wild card.
"He had some chances at the beginning but I found my game," Almagro said. "Winning the first match in straight sets, in a very short time only helps, especially with all the heat."
Almagro will play Potito Starace of Italy next.
Three other Spaniards also started well, with two-time champion Carlos Moya fighting past countryman Pablo Andujar 6-4, 6-7 (2), 6-2; seventh-seeded Marcel Granollers topping Fabio Fognini of Italy 7-6 (3), 7-5; and Alberto Martin putting away Marcos Daniel of Brazil 6-4, 7-5.
Second-seeded Gael Monfils celebrated his career-high No. 10 ranking this week by firing 17 aces on the clay past fellow Frenchman Olivier Patience in a 7-6 (4), 6-4 win in his Acapulco debut. Thomaz Bellucci of Brazil was next.
Fifth-seeded Jose Acasuso won an all-Argentine match with Brian Dabul 6-3, 7-5.
Women's second seed Pennetta swept Mariya Koryttseva of Ukraine 6-3, 6-1 to improve her record in the tournament to 25-4. Since she reached the quarterfinals on debut in 2003, Pennetta has made it to the final each year, winning in 2005 and last year. Of her six career titles, four have been in Latin America.
She faces Pauline Parmentier of France next.
Third-seeded Carla Suarez Navarro of Spain rebounded from the left elbow injury she sustained in Bogota by defeating qualifier Viktoriya Kutuzova of Ukraine 7-5, 6-3.
Other winners included lucky loser Agnes Szavay of Hungary, Barbora Zahlavova Strycova of the Czech Republic, Mathilde Johansson of France and Maret Ani of Estonia, who beat Bogota semifinalist Patricia Mayr of Austria 4-6, 7-6 (1), 6-4.