The second-seeded defending champ dropped the first set but came back for a 4-6, 6-3, 7-6 (7-4) victory over Starace in a 2-hour, 7-minute match at Buenos Aires Lawn Tennis Club.
The Spaniard continued his torrid play after defeating Italy's Filippo Volandri for the crown of last week's Brazil Open. He claimed last year's Buenos Aires title with a win over Juan Ignacio Chela.
Almagro, the world No. 11, now moves on to face Argentine Federico Delbonis.
Also advancing Tuesday was No. 3 seed Gilles Simon of France, who got past Portuguese Rui Machado 6-3, 7-5. No. 5 Juan Monaco of Argentina also moved on with a 6-1, 3-1 win when Filippo Volandri of Italy retired.
Reindeer recovered after escaping from Santa during lighting ceremony
Wisconsin business offering 'therapeutic cuddling' forced to close