Palermo's penalty attempt flew over the crossbar, but he atoned in the 60th minute with a header to make it 2-0.
Rodriguez completed the scoring with a solo run.
Caracas had barely squeezed into the last 16 but twice rallied to draw with the two-time champions Santos.
They had won all eight of their Copa matches this year but were under pressure after losing 2-0 to Sao Caetano last weekend in their state finals in Brazil.
Ze Roberto of Santos took advantage of a blunder by Jorge Rojas to score in the 17th minute but Venezuela's Caracas equalized when Ivan Velasquez scored from a free kick.
Kleber put Santos back in front, but the Venezuelans struck back four minutes from the end with a goal from defender Leonel Vielma.
Gremio started well but the host's and three-time champion's Sao Paulo finished better.
After a scoreless first half, Sao Paulo turned up the pressure and were rewarded in the 58th when Miranda blasted into the left corner for the only goal.
Minutes later, Souza set up Dagoberto who whipped around and volleyed the shot home but the goal disallowed for offside.
Gremio coach Mano Menezes introduced former AC Milan and Sao Paulo striker Amoroso in the second half but the 32-year-old striker had little impact.
Mexico's America scored twice in a three-minute span in the first half and ended up routing Chile's Colo Colo 3-0.
Defensor Sporting thrashed Flamengo of Brazil 3-0 in Montevideo, with Alvaro Navarro scoring in the fifth minute from a penalty kick and again in the 75th, in between Alvaro Gonzalez's header goal three minutes after halftime.
Navarro's opening goal followed Flamengo defender Leo Moura's foul on Defensor striker Paulo Pezzolano before 10,000 fans at Centenario stadium.
On Thursday, Parana of Brazil take on Libertad of Paraguay, Nacional of Uruguay play Necaxa of Mexico, and Cucuta Deportivo of Colombia face off against Toluca of Mexico.