Luiz Felipe Scolari's men earned a 2-1 win to continue their run in front of their home faithful, although the Colombians were aggrieved by decisions which seemed to favour the hosts.
Monaco man Rodriguez is the second youngest player to score six goals at a World Cup finals behind Pele's effort in 1958, and the 22-year-old was visibly emotional at the final whistle in Fortaleza on Friday.
"We fulfilled all the dreams we had… it's a shame we have to go," Rodriguez said.
"I'm crying because we gave everything. Unfortunately, the referee didn't help much.
"Brazil is a good team but we gave everything so we are calm. We wanted to carry on, but we hold our heads high."
Rodriguez paid tribute to the nation's fans, and said the playing group were not ashamed of their last-eight exit.
"Thank you Colombia, because you always had faith," he said.
"Unfortunately we stop here, but we wanted to carry on. We're sad but we also have to feel proud because we left our skin out there."
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