In a match Colombia had to win to survive the group stages, the South Americans' patience paid off with Yerry Mina breaking the deadlock in the 40th minute with a powerful header from a superbly-curled James Rodriguez cross that gave the big Barcelona defender his maiden World Cup goal.
Robert Lewandowski said after Poland were eliminated from the World Cup with a 3-0 defeat to Colombia that his side lacked the quality needed to progress, admitting "it's not like we lost by a whisker".
The chance for a World Cup goal has been a long time coming for Radamel Falcao.
Yerry Mina, Radamel Falcao and Juan Cuadrado were all on the scoresheet in a convincing win for Colombia in Kazan.
A 2-1 defeat to Senegal in their Group H opener has left Poland in desperate need of their first three points of the tournament at Kazan Arena.
The victory puts Colombia in third place in the group on three points behind the Japanese and Senegal, who shared a 2-2 draw earlier on Sunday to move to four points apiece.
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Poland struggled to cope with Colombia's midfield threats, with Juan Quintero starring alongside Cuadrado and Rodriguez.
The pressure is now on, not just for Rodriguez, but in the knowledge that Jose Nestor Pekerman's six-and-a-half-year stint - only three other coaches at the World Cup have been in place longer - in charge would have a huge stain should Colombia lose against a Poland side ranked eighth by Federation Internationale de Football Association and also needing a result.
"I'm very sorry and very sad about the result, but tomorrow's another day and we have to come to terms with things", he said. "So we fought, we tried to do our best, but at the moment we could not do anything more", the prolific Bayern Munich striker told Polish TV.
James Rodriguez played in Falcao, who sprinted through on goal and then tried to set up Cuadrado with a clever backheel, but got his execution all wrong. The lively Quintero threaded a defence-splitting pass to Falcao, who rolled back the years with a delightful low finish with the outside of his right boot.
Poland faces Japan, and Colombia plays against Senegal, on Thursday.
Robert Lewandowski, making his World Cup debut in Russian Federation, struggled with a lack of service throughout, the Bayern Munich striker going closest in the final few minutes when a long-range effort was tipped over by David Ospina.