Delbonis, ranked 61 in the world, was the only seeded player on the clay court Tuesday and topped Cuevas 6-3, 6-4.
Aljaz Bedene of Slovenia ousted Stephane Robert of France 2-6, 6-3, 6-3 and moved on to a second-round clash with top-seeded Fabio Fognini of Italy.
Other winners Tuesday included Taro Daniel of Japan, who eliminated 2010 Chile Open champ Thomaz Bellucci of Brazil.
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