The striker, who is currently on international duty with Colombia at the Copa America, had appeared set to join AC Milan before Atletico registered their interest.
"It was finalised today," Luiz Henrique Pompeo told AS. "We want to thank Milan for their interest right up the end, we feel very honoured by it, but you can't be in two places at once.
"Atleti are a very big club that has done important things in the league and in Europe. Jackson is going there to also achieve great things and to bring the best out of himself.
"That's his dream and he's very excited. He's really happy."
Martinez will become the second striker Atletico have signed this summer, following the arrival of Luciano Vietto from Villarreal.
The 28-year-old joined Porto from Chiapas in 2012 and scored 21 league goals last season.