The Brazilian side took an early lead and were comfortable from then on, with Damiao to the fore in a show of excellent football.
Dagoberto opened the scoring in the fourth minute of the game after receiving the ball from Oscar, before Damiao made it 2-0 just three minutes later with a header.
Twelve minutes after the break, Damiao scored his second before he completed his hat-trick in the 73rd minute following another assist by Oscar.
Jo came on for Damiao seven minutes later and he needed only 60 seconds to get on the scoresheet himself.
The win took Inter joint top of the group with their opponents, with both sides having collected six points from the opening three games.
Damiao told reporters in Porto Alegre he was unimpressed with the defence of the Bolivian side.
"They did not know how to mark well," he said. "Our midfielders found me very well to assist me.
"I only work to help the team. Goals are important, but not more than a victory for the team."
In Group Two duel in Argentina, Mariano Pavone gave Lanus a 1-0 home win over Ecuadorian side Emelec.
Pavone finished a Mario Regueiro's cross from the right in the 71st minute and gave Gabriel Schurrer's side the win which saw them share top position with Brazilian side Flamengo.
Nacional of Paraguay recovered from two defeats to claim a 3-2 triumph over Venezuela's Deportivo Tachira.
Pedro Fernandez put Tachira ahead in the 31st minute, but a 50th-minute penalty gave the hosts the equaliser.
In the 72nd minute, Cano scored his second goal and, even though Angel Chourio levelled matters again in the 79th, Silvio Torales made sure the Nacional fans could celebrate.
In Peru, a 16th-minute header from Jose Fernandez-Piedra helped Alianza Lima to a 1-0 win over Uruguayan giants Nacional, who ended the game with nine men.
Midfielder Israel Damonte and defender Diego Placente were sent off during the second half.
It was Alianza Lima's first win in this year's Libertadores.
Finally, an 11th-minute header by Carlos Fierro gave Mexican giants Guadalajara a 1-0 win over Defensor Sporting of Uruguay.