The Brazilian Serie A side thrashed Once Caldas 4-0, but the hosts in Sao Paulo finished the contest with nine men.
Tachira, of Venezuela, were 2-1 winners over Cerro Porteno in San Cristobal.
Corinthians got their continental campaign off to an ideal start, leading inside the first minute via Emerson Sheik.
The former Qatar international curled what looked to be an over-hit cross into the back of the net to send the Arena Corinthians faithful into raptures inside 30 seconds.
The hosts would end the half less-optimistic of a positive result, however, as they were down to 10 men when striker Paolo Guerrero was dismissed in the 27th minute for a swinging arm on Camilo Perez.
But in the 10th minute after the half-time interval, Corinthians doubled their advantage when Felipe nodded home a Jadson corner.
Both sides were down to 10 men in the 69th minute, when Luis Murillo earned a second yellow card for a raised arm.
Midfielder Elias added Corinthians' third goal in fine style, the Brazilian playing a one-two-three with his team-mates before finishing by lifting the ball over goalkeeper Jose Cuadrado.
Fagner made it four in the 79th minute, before Fabio Santos was handed a straight red card in the 91st minute to slightly sour Corinthians' runaway victory.
In Venezuela, the hosts got off to a flying start - Javier Lopez and Jorge Rojas struck seven minutes apart to put Daniel Farias' men clear early.
Jonathan Fabbro scored what could be a vital away goal for the Paraguayan visitors, netting in the 50th minute to give his side hope heading to the second leg on February 11.
