The fifth-seeded Zeballos defeated seven-time French Open champion Rafael Nadal in the final of the VTR Open in Chile just over a week ago.
Expectations were high for him playing at home.
After winning the second-set tiebreaker, Bedene then broke Zeballos in the second game of the deciding set en route to the victory.
"It was my first match since August, I think, on clay. So at the beginning I was confused," Bedene said. "I'm really happy with my performance -- first round, first win."
Bedene apologized for beating the hometown favorite at the Buenos Aires Lawn Tennis Club.
"Horacio is my good friend, so I feel sorry for him," Bedene said. "But tennis is tennis and somebody has to win."
In 2012, Almagro won on clay in Brazil and was the runner-up in Buenos Aires. In 2011, the Spaniard won twice in Latin America -- in Brazil and Buenos Aires -- and was the runner-up in Acapulco, Mexico.
"This trip through Latin America has been very good for me in recent years," Almagro said. "But every year is different."
In other first-round matches, No. 4 seed Thomaz Bellucci of Brazil was ousted by Argentine Diego Schwartzman 6-4, 4-6, 6-1, eighth-seeded Albert Ramos of Spain advanced with a 6-1, 6-0 win over Flavio Cipolla of Italy and Federico Delbonis of Argentina moved on with a 6-1, 6-4 victory over Portugal's Joao Sousa.