Montanes broke Fognini four times in the match, including the only break of the third set, and won 50 percent of the points on the Italian's second serve.
"I'm really happy because the match was really, really tough," the Swiss said. "Victor is a really strong player, so it's never easy to play against him."
In a late match, No. 5 seed Juan Monaco defeated fellow Argentine Eduardo Schwank 7-6 (5), 6-2 to advance to the quarterfinals. Earlier, eighth-seeded Juan Ignacio Chela of Argentina defeated countryman Horacio Zeballos 7-5, 6-2.