The late victory, England's second ever at the World Cup after conceding the first goal, earned the 1966 champions a spot in the round of 16 and a match against co-hosts Mexico at their famed Azeca Stadium.
"It was just about pounding the rock, keep pounding the rock and our moment would come," Kane said after scoring his fourth and fifth goals of the tournament.
"We spoke about people having hero moments. It can be anyone in the team … Whoever it is, we have hero moments, and for me it was the day."
Getting there didn't come easy against a Congo team who nearly pulled off one of the biggest upsets in World Cup history.
England's only other World Cup victory after trailing 1-0 came in the 1966 final against West Germany at Wembley Stadium.
Kane made sure it happened again by extending his record as his country's leading scorer in tournament history to 13. He has record 84 goals for the national team in total.
His first goal on Wednesday came in the 75th minute. Substitute Anthony Gordon lifted a cross from the left and despite Congo goalkeeper Lionel Mpasi getting a hand to Kane's header, he could not stop it from nestling into the bottom corner.
Eleven minutes later, Kane sent his shot ripping into the top corner to put the thoughts of an upset to rest.
"When you get to the knockouts, the pressure and the risks are so much higher. But from an attacking point of view, that was easily our best game of the tournament," Kane said.
"Sometimes you just have to grind wins out and we did exactly that today. I told the boys to enjoy it. We're through, and we go again in four days."
Brian Cipenga had put Congo in the lead in the seventh minute after collecting a cross on the left of the box and shooting low past England goalkeeper Jordan Pickford at the near post.
"We're disappointed because we really believed we could do it," Congo coach Sébastien Desabre said.
"We played well. Towards the end of the match, we conceded two chances and one of the world's best players scored two goals against us. It's a shame."
