Monaco striker Falcao stroked home his 30th international 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.
Barcelona defender Yerry Mina scored the opener, powering a header beyond Wojciech Szczesny five minutes before half-time to enliven the contest, and it was the creativity and industry of Juan Cuadrado and James Rodriguez that picked apart Poland's tightly stitched seams.
The presence of James, fully fit again after a calf injury, vastly improved Colombia's link-up play after a below-par performance in its opening 2-1 defeat to Japan, and he provided the assist for Cuadrado to add a third goal that put the game well beyond Poland.
Poland will have a role to play in deciding the fate of the group when it takes on Japan, but ultimately coach Adam Nawalka can have few complaints about being dumped out of the competition after two games in which his side failed to provide talismanic striker Robert Lewandowski with adequate service.
The first ever meeting between these two sides began with a number of heavy challenges at both ends of the field that prevented the game from developing into an early rhythm.
Colombia started to dominate the midfield and launched attacks down the right flank through Cuadrado, but the Juventus winger failed to penetrate a packed Poland defence.
Nawalka's side set its stall out to avoid falling behind as it did in the first half of its 2-1 defeat to Senegal and Colombia's chances of breaking its resolve were hit by an injury to Abel Aguilar, who was taken from the field on a stretcher in the 32nd minute and replaced by Mateus Uribe.
Shortly after Cuadrado came alive, running at the Poland defence and starting a move that culminated with an exquisitely weighted cross by James that was met by the head of Mina, who lost his marker to head over Szczesny's outstretched arms and into the net.
Juan 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.
The isolated Lewandowski's first real chance of the game arrived in the 58th minute when he controlled a long pass brilliantly only to shoot straight into David Ospina's midriff as Poland attempted its first shot on target of the game.
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.
With 15 minutes left on the clock, James's pass from the left found Cuadrado in space through the middle and he took a touch before applying an assured finish into the bottom-right corner of the net.
Colombia almost added a fourth after a mix-up between Jan Bednarek and Szczesny, but Uribe's backheel was cleared off the goal line to leave the final score 3-0.