Rodriguez netted his sixth goal of the finals during Friday’s clash with Brazil, who progressed to the semi-finals with a 2-1 victory in Fortaleza.
Thiago Silva opened the scoring for the tournament hosts after just seven minutes – following a corner – before David Luiz doubled the margin with a stunning long-range free-kick on 69, with Rodriguez pulling a goal back from the penalty spot 12 minutes from time.
Colombia won all three round-robin games to top Group C and the 22-year-old Rodriguez said: "We are sad but very proud.
"The referee was wrong but Brazil is very good. David Luiz and Daniel Alves are great players.
"We wanted to carry on but we hold our heads high. Thank you Colombia. We're sad but we also have to feel proud because we left our skins out there."
Colombia's head coach Jose Pekerman said: "We dreamed of this for James Rodriguez. He was one of the protagonists of the World Cup.
"I think that in the analysis of the match, the early goal for Brazil made a big difference because we are not used to conceding goals like this and that drained us of confidence a bit.
"Going ahead so early calmed them down and that was very important.
"After being absent from the World Cup for a long time the performance here has made Colombian football valued.
"We have shown we have some very talented players and also competed with an important spirit."