The 27th-ranked Cuevas needed less than 90 minutes to beat No. 67 Carreno Busta, who was playing in his first ATP final.
Last Sunday, Cuevas won the Rio Open after beating Rafael Nadal in the semifinals.
"This crowd makes me feel at home in Brazil," Cuevas said.
Four trained shelter dogs who retrieved balls that went out of bounds in an exhibition match on Thursday were also present for the final.