Defender Yerry Mina scored the opener in the 40th minute with captain Radamel Falcao and winger Juan Cuadrado putting the gloss on the Group H win with two goals in five minutes in the last quarter as the desperate Poles chased the game.
Midfielder James Rodriguez, winner of the Golden Boot at the last World Cup, returned to the starting lineup after injury and played a part in two of the goals as Colombia got back on track after their opening loss to Japan.
The victory puts the South Americans in third place in Group H on three points behind Japan and Senegal, who shared a 2-2 draw earlier on Sunday to move to four points apiece.
A Colombia victory over the Africans in their final first-round match in Samara on Thursday would secure a spot in the last 16 and keep them on target to at least match their charge to the quarter-finals in Brazil four years ago.
For Poland, the optimism fostered by their quarter-final spot at Euro 2016 and a strong qualifying campaign proved to be misplaced as prolific striker Robert Lewandowski drew a blank and they slumped to a second successive defeat.
The match started to a barrage of noise from the yellow-shirted Colombian fans on a sweltering evening at the Kazan Arena and the early exchanges were frenetic with both sides keen to make up for their opening losses.
Poland’s offensive tactics were limited to launching long balls toward Lewandowski and although Colombia were trying to play the ball on the floor, they could not get their passes to stick.
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.
The respite was brief and on the next Colombia attack Juan Quintero found Rodriguez inside the box and he chipped the ball into the danger area for Mina to head into the net from point-blank range.
The Poles came out firing at the start of the second half and Lewandowski finally got a chance to open his World Cup account in the 58th minute but his first touch on the long ball was not his best and goalkeeper David Ospina smothered his shot.
Colombia saw off the danger and doubled the lead after 70 minutes when Quintero’s pass allowed Falcao to beat the last defender before stroking the ball past Szczesny with the outside of his boot.
Poland continued to push forward and five minutes later they were caught on the break when Rodriguez rasped a pass across the field to set Cuadrado free on goal and allow the winger to claim fitting reward for a fine performance with a cool finish.
Lewandowski had a thunderous drive tipped over the bar by Ospina and the keeper put his body on the line again to block a Dawid Kownacki shot from close range but it was too little, too late.
