Gareth Southgate, a villain in England's Euro '96 shootout agony, seemed poised for more penalty woes when David Ospina saved from Jordan Henderson, only for Jordan Pickford and Eric Dier to emerge as national heroes.
Six of England's eight previous goals at this tournament had come from dead-ball situations but a series of early set pieces were well defended by Colombia.
Kieran Trippier's delivery from open play was as dangerous as usual and he provided the first chance of the night for Kane, who, leaning back, could not direct a close-range header underneath the crossbar.
Referee Mark Geiger missed the incident, which was picked up by the VAR officials who instructed the American to issue only a yellow card despite it being clear that Barrios thumped his head into the Liverpool captain's chest.
That episode sparked a lively end to the first period as Trippier and Kane shot wide from outside of the box, and Juan Quintero had Colombia's first effort on target with a 25-yard strike which Pickford held with authority.
Colombia's bullying tactics escalated after the break and they were punished on 54 minutes when Carlos Sanchez, who was sent off in his country's opening Group H match, was called up for wrestling down Kane from a corner.
While all about him were losing theirs, the England captain kept his composure following a three-minute delay to slot down the middle from 12 yards for his sixth goal of the tournament - two more than his nearest rival.
The encounter threatened to spiral out of control as Colombian tempers flared. England largely kept their cool although John Stones was lucky to get away with what could have been interpreted as a deliberate kick to Falcao's head.
Colombia collected five yellow cards before the 70th minute and their dirty tricks were at the detriment of their attacking play as they only seemed to focus their energy on grabbing an equaliser in the final 10 minutes.
Kyle Walker's complaceny in his own half allowed Juan Cuadrado a golden opportunity on 82 minutes. The Juventus winger was sent clear by Carlos Bacca but leant back to slice a wild shot into the stands from 15 yards out.
Falcao headed over the crossbar from Johan Mojica's delivery and when Pickford acrobatically pushed Mateus Uribe's speculative 40-yard volley behind with his fingertips, England's spot in the quarter-finals looked safe.
However, towering defender Mina rose from a stoppage-time corner to head home his third goal in three games, squeezing his effort between Pickford and the defender on the line to devastate England and keep their own tournament alive.
The momentum was against Southgate's men and Colombia spent all of the first half of extra time on the front foot, creating one half-chance for Falcao when Harry Maguire's pressure forced him to head wide of the near post.
England showed admirable character to have the better of the second additional period. Substitutes Danny Rose and Dier went close, the former with a left-footed shot from Henderson's through-ball, and the latter wasting a free header.
For the fourth time at a World Cup, England's fate would be decided by a penalty shootout and the first five attempts were all scored - by Falcao, Cuadrado and Luis Muriel for Colombia and by Kane and Marcus Rashford for the Three Lions.
Ospina dived low to his left to keep out Henderson, only for Uribe to smash Colombia's fourth against the crossbar. Trippier kept his nerve before Pickford's save from Bacca set up Dier to win the game and send England into the last eight.
Sweden await next for Southgate's boys whose extra belief at finally shaking their shootout hoodoo could be enough to take them to the World Cup semi-finals.
COLOMBIA (4-3-2-1): Ospina; Arias (Zapata, 116'), D. Sanchez, Mina, Mojica; Lerma (Bacca, 61'), Barrios, C. Sanchez (Uribe, 79'); Cuadrado, J. Quintero (Muriel 88'); Falcao
ENGLAND (3-4-2-1): Pickford; Stones, Maguire, Walker (Rashford, 113'); Trippier, Young (Rose, 102'), Lingard, Henderson; Dele (Dier, 81'), Sterling (Vardy, 88'); Kane