In the 75 minute, the match was over as a contest when Rodriguez played Radamel Falcao through on goal and he made no mistake with just the Polish keeper to beat, 3-0. Barcelona star Yerry Mina scored the opening goal for Colombia with a powerful header before Radamel Falcao bagged his first World Cup goal, which was followed by a brilliant finish from Cuadrado who burst his lungs to race past a couple of Polish defenders before finishing off a stunning effort.
Earlier on Sunday, Senegal and Japan, the other two countries in Group H, tied 2-2.
Mina out-jumped the Poland defense to head in the opening goal from James Rodriguez's cross, rewarding a spell of domination by the Colombians.
Colombia did well to shut down the striker with two players always close on his heels, and Poland did not have the quality or the skill to switch their tactics.
The 29-year-old striker has averaged nearly a goal a game for Bayern in the past three seasons, and 57 percent of Poland's goals in World Cup qualifying came from the Warsaw native.
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Former England global Gary Neville has lavished praise on striker Harry Kane after his hat-trick in the World Cup on Sunday. Only one English player - Lineker with 10 - now has more World Cup goals than Kane, who has swiftly moved to five goals.
Juan Quintero played a lovely through ball to Falcao and he did the rest in the 70th minute.
"I'm very sorry and very sad about the result, but tomorrow's another day and we have to come to terms with things", said Poland coach Adam Nawalka.
Poland began to fade as the first half wore on and some standout play from Juan Fernando Quintero and Cuadrado set Santiago Arias away into space on the right, but the full-back saw his cross turned away by Michal Pazdan at the near post.
Rodriguez has now been directly involved in nine goals in seven World Cup appearances for Colombia, scoring six and assisting on three. "I want, and we want, to share it with him", he said.
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.