After a barren first half, Mina (74th minute) headed in a corner to send Colombian players and supporters into delirium as they needed nothing less than a victory to secure qualification.
Colombia finished on top of the group with six points, while Senegal finished level on four points with Japan, who lost 0-1 to already-eliminated Poland in the other group game played simultaneously.
Both Japan and Senegal had even goal differences and had scored four goals each, but Japan advanced as runners-up because they only picked up four yellow cards in the group stage, while Senegal had six.
This is the third time Colombia have reached the last 16. This time, it boiled down to their final group game in which chances were few and far between as both sides started cautiously keeping in mind the stakes of the game.
Colombia forward Radamel Falcao won a free kick in the 11th minute while Juan Quintero’s shot was saved by the Senegal goalkeeper.
Senegal were then denied a penalty in the 18th minute after a VAR review in the 17th minute.
Sadio Mane was brought down by Colombia defender Davinson Sanchez inside the box and Senegal was awarded a spot kick. Following the VAR check, the referee reversed his decision and cancelled the penalty for the Teranga Lions afterwards.
In the 31st minute, the South American side suffered a major jolt when James Rodriguez, 2014 edition’s top scorer, was taken out by coach Jose Peckerman in what looked like an injury.
In the second half, Colombia looked to up the ante but Senegal too put pressure and goalkeeper David Ospina was forced into some good saves to keep them at bay.
There were number of chances wasted at both ends too as time ticked on.
Then came Colombia’s moment of reckoning when Quintero’s corner was majestically met by Mina to bullet home a downward header and send their swathes of supporters into raptures.
Senegal tried hard to stage a comeback but their main man Mane looked off colour as they eventually failed to pull one back.
With Senegal’s exit, this will be the first time since the Round-of-16 came into being in 1986 that there will be no African teams represented.
Meanwhile in Volgograd, a Jan Bednarek strike in the 59th minute handed Poland a triumphant exit from the World Cup.
Poland have three points from three games and are in the fourth and final spot in the group.
Despite the importance of the game, Japan coach Akira Nishino made six changes to his starting 11 after repeating the same team for the first two matches, while Lukasz Fabianski and Kamil Grosicki were in the Polish side.
Grosicki’s pace almost created a chance for Robert Lewandowski in the 5th minute, but the striker’s touch let him down in the Japan penalty area, reported Xinhua news agency.
Japan took time to get into the game, but Shinji Okazaki headed wide in the 12th minute, Yoshinori Muto produced a smart save from Fabianski a minute later and Maya Yoshida also worked the keeper from the edge of the area.
Japan were leaving spaces in midfield and were almost punished in the 32nd minute when Grosicki’s header drew a good save from Eiji Kawashima in the Japan goal, although Japan responded quickly as Takashi Usami drew another stop from Fabianski.
There was a setback for the Japanese at the start of the second half when Okazaki had to be replaced by Yuya Osako after less than two minutes with what looked like a muscle problem.
However, Japan retained their attacking 4-4-2 formation with both wide midfielders pushing forward, almost allowing Grosicki and Lewandowski to combine in the 54th minute.
The game again settled into a pattern with Japan looking to attack, but never quite in control and were punished just before the hour when Jan Bednarek side-footed home a free kick from close range, getting the better of his Southampton teammate Yoshida in the process.
With Japan now needing a goal to qualify for the next round Takashi Inui replaced Usami, but they should have fallen two behind when Lewandowski fired another Grosicki ball over in the 74th minute and the Hull City winger almost forced Tomashi Makino to concede an own goal with another dangerous cross.
News of Colombia’s goal against Senegal clearly filtered through and with the result in the other Group H game meaning Japan would qualify through having fewer yellow cards than their rivals both sides appeared to draw a footballing truce for what was frankly an embarrassing last 10 minutes in which neither tried to score.