Monaco striker Falcao stroked home his 30th global goal, making him the first player from Colombia to reach such a milestone, midway through the second half of a cagey encounter in Kazan, where defeat for either side would have meant elimination from the competition. "I'm happy for the victory, for the team's performance and the goal that I scored, which I've been waiting for for many years".
The attacking triumvirate of Rodriguez, Cuadrado and Juan Fernando Quintero were all involved from a short corner and the former was allowed space to deliver a ideal cross from the right that Mina just had to guide beyond an oncoming Szczesny with his head.
The loss signalled the end of Poland's hopes of qualification for the last 16 after two losses from their opening two matches, while the Colombians enter the final round of group matches, against Senegal, with their fate in their own hands.
"I see that it is very hard for them to fall into that role, because they have never done it, we are mentally well, but we have to improve because we are ambitious, we came with that intention".
"He is a symbol of the national team".
Having survived that scare, Colombia came to life and Quintero's superb slide-rule pass presented Falcao with a one-on-one situation and he expertly guided the ball past Szczesny with the outside of his right foot to double his side's lead.
Cuadrado was causing some problems down Poland's right flank, however, and a piece of mazy dribbling in the 37th minute saw the winger forge the first genuine opening, which goalkeeper Wojciech Szczesny blocked behind.
Rodriguez's through ball after Poland lost possession in midfield led to the final goal.
Juan Fernando Quintero fired wide from 25 yards as Colombia started the second half brightly and Cuadrado then squared to Falcao on the edge of the box, but the Monaco striker lifted his shot well over the crossbar. Quintero came close in the 56th minute on the break, but his goal-bound shot took a deflection.
The win for Colombia sees them on three points in Group H, placed third behind Japan and Senegal, who have four points each.
Cut off from the rest of his teammates, Robert Lewandowski had a few attempts but could not convert his rare chances.
Given his first start after fully recovering from a calf injury, Rodriguez lived up to expectations. He formed an excellent partnership with Quintero to create some lovely play for Colombia. "But fighting is not enough to win World Cup matches, you also have to have quality, and we had too little of that".
Poland's offensive tactics were limited to launching long balls towards their prolific striker and although Colombia were trying to play the ball on the floor they could not get their passes to stick.
Poland has now failed to keep a clean sheet in its last 10 World Cup matches, since beating Portugal 1-0 in 1986.
"We know what James can do, he has played for many years at the top level in the world", Poland midfielder Piotr Zielinski said.