In a match destined to see the loser sent home after the group stages, Colombia overcame a nervous start against Adam Nawalka’s men to move up to third place in Group H, one point behind Japan and Senegal.
The disappointing Poland became the first European side eliminated from the tournament.
The South Americans’ bid for a place in the last 16 will be determined by the results from their final game with Senegal, who drew 2-2 with Japan earlier in the day.
Colombia’s march to the quarter-finals in Brazil four years ago came thanks to James Rodriguez’s six-goal tally for ‘Los Cafeteros’.
And days after a troublesome calf injury restricted him to a half-hour cameo role as 10-man Colombia stumbled to a 2-1 defeat against Japan, the Bayern Munich midfielder was back to his best.
Starting alongside Juan Quintero and Cuadrado on the right, Colombia’s ultra-offensive starting line-up was a forewarning.
After riding a tense start that forced Mina and then Wilmar Barrios to react quickly to thwart early threats from Robert Lewandowski, Colombia eventually got into their stride.
Cuadrado was wasteful on several occasions when finding space deep on the right flank.
The Juventus winger’s trickery would eventually pay dividends, but it was Falcao – who has dreamed all his life of scoring a World Cup goal – who turned on the style, dancing through the Polish defence to win the corner that led to Colombia’s opener.
Rodriguez’s short corner eventually found Quintero, whose smart through ball back to Rodriguez gave him time and space to curl a perfect left-footed cross that Mina rose to header powerfully past Szczesny on 40 minutes.
Colombia resumed in positive fashion, Cuadrado holding up well to set up Falcao for a first-time drive that sailed over Szczesny’s crossbar.
A Colombia counter saw Falcao’s drive from the edge of the area deflected out for a corner.
Nawalka replaced Dawid Kownacki with the more experienced Kamil Grosicki and the switch almost paid dividends.
Only the bravery of goalkeeper David Ospina saved Colombia from conceding what looked like the equaliser when he rushed out to block from Lewandowski after the Bayern Munich striker did well to get a shot away from a long ball from midfield.
Poland threatened again, but when Lewandowski got his head to a cross into the area he was marked by both Mina and Davinson Sanchez.
Colombia, however, were not to be denied.
When right-back Sanitago Arias found Quintero, the midfielder sliced Poland’s defence apart with a low pass that found Falcao on the run before he beat Szczesny down low with the outside of his right boot.
Five minutes later, Rodriguez produced arguably the assist of the night, drawing several Polish players on the left flank before somehow finding Cuadrado on the run with a superb cross-field pass that the pacey winger tucked away.
World Cup 2018: Japan come back twice against Senegal to draw 2-2
Japan and Senegal remain well placed to reach the knockout stage of the World Cup – and a last-16 date with England or Belgium – after an entertaining draw in Ekaterinburg.
Sadio Mane put Senegal ahead from close range after an awful mistake by Eiji Kawashima, the ball flying past the Japan keeper after his attempted punched clearance hit the Liverpool forward.
Japan were level before half-time through Takashi Inui’s lovely curling finish, before Yuya Osako hit the crossbar.
Nineteen-year-old defender Moussa Wague thought he had sealed Senegal’s second straight Group H win with a fine finish from an angle but substitute Keisuke Honda salvaged a point for Japan moments after coming on.
Japan and Senegal have four points after two games, while Group H rivals Poland and Colombia are both chasing their first points of the tournament when they meet in Kazan at 19:00 BST.
Despite another point, Senegal will be disappointed they could not follow up their 2-1 win over Poland last week by becoming the first African side to win two group games since Ghana in 2006.
But 16 years on from their debut World Cup, the Teranga Lions continue to win new fans in Russia.
While Egypt, Morocco and Tunisia are already making plans to return home after failing to secure a point, Senegal can still ensure Africa has at least one representative in the last 16.
However, Aliou Cisse will be frustrated his players failed to build on Mane’s opener after Kawashima’s howler.
Senegal were well in control but failed to build on their strong start and Inui’s smart finish, after Yuto Nagatomo created the chance, saw Japan draw level with their first shot on target.
Even after Wague restored Senegal’s advantage with a sweeping finish, they were unable to see the game out – keeper Khadim N’Diaye’s failed clearance leading to Honda’s equaliser.
In 2002, the likes of Henri Camara and El Hadji Diouf helped Senegal reach the quarter-finals under Bruno Metsu. If Senegal’s class of 2018 hope to emulate that achievement they will have to cut out the mistakes.
England hat-trick hero Kane basks in World Cup rout
Hat-trick hero Harry Kane says “it´s all coming together” for England at the World Cup after Gareth Southgate´s side sailed into the last 16 with a 6-1 rout of Panama on Sunday.
Kane´s treble and two goals from John Stones, with Jesse Lingard also on the scoresheet, meant England joined Belgium on six points at the top of Group G ahead of the match between the two teams on Thursday.
Kane said: “I´m very proud. Not many players get to score a hat-trick in the World Cup.
“We have to enjoy this. It´s not every day we get to go through after two games.”
He said his third goal, which came from a Ruben Loftus-Cheek shot that deflected off the English captain´s heel into the net, “was one of the luckiest of my career”.
His five goals took him to the top of the World Cup Golden Boot scoring chart with five goals in two games ahead of the four scored by Cristiano Ronaldo and Romelu Lukaku.
“It´s going well, there´s still a long way to go, the most important thing is winning games,” Kane said.
“There´s a lot of good players behind me, hopefully I can take it into Thursday´s game.”
Earlier, Kane told BBC TV he was “proud of the boys”.
“We knew it was going to be tough but it´s all coming together. It´s a brilliant result, fantastic to be through and to do it in this way is great.”
The Tottenham Hotspur striker said England fans were justified in dreaming of winning the World Cup for the first time since 1966.
“We´re not getting too ahead of ourselves. We´ve got to keep it going but, yes, you have to believe. If you want to achieve anything in life you have to believe.
“We´ve got to stick together, stick to our game plan.”