Thomas Tuchel's side were given a huge scare by DR Congo, who scored through Brian Cipenga seven minutes into the first World Cup knockout game in their history, but Kane came to the rescue.
After heading in substitute Anthony Gordon's cross to bring England level with a quarter of an hour remaining, Kane rifled home a stunning 86th-minute winner to complete the comeback.
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The double takes Kane above Brazil legend Pele on 13 goals at World Cups and ensures England will face Mexico at the Azteca Stadium in the last 16, kicking off at 1am (BST) on Monday.
England were made to work hard for the victory, with Cipenga capitalising on defensive disarray to beat Jordan Pickford at his near post for the shock opener before DR Congo goalkeeper Lionel Mpasi made a string of saves to keep England at bay at the other end.
Jude Bellingham was thwarted on three occasions, while Marcus Rashford had a shot cleared off the line by Aaron Wan-Bissaka, leaving it to Kane to make the breakthrough having earlier been controversially denied a penalty after going to ground following contact from Mpasi inside the box.
England laboured at times against their disciplined opponent, but Tuchel's changes made a difference and Kane's emphatic winner, fired into the roof of the net after finding space amid a crowd of defenders, kept their World Cup dream alive.
The key moments from Atlanta...
- 7: GOAL! Brian Cipenga beats Jordan Pickford at his near post as DR Congo score shock opener
- 30: SAVE! England finally have their first shot as Jude Bellingham's header is palmed away by Lionel Mpasi
- 35: OFF THE LINE! Marcus Rashford has a shot cleared off the line from Aaron Wan-Bissaka
- 42: OFF THE POST! Let-off for England as Yoane Wissa hits the post from close range
- 43: NO PENALTY! Harry Kane is controversially penalised for diving after appearing to be caught by Mpasi in the box
- 46+2: SAVE! Bellingham is denied again as Mpasi saves his header from Noni Madueke's cross
- 45+6: SAVE Mpasi keeps out Harry Kane's shot at his near post
- 53: SAVE! Mpasi rescues DR Congo again as Bellingham's deflected cross is saved
- 75: BREAKTHROUGH! Kane heads in England's leveller from Anthony Gordon's cross
- 86: GOAL! Kane makes space for himself in the Congo box and rifles an unstoppable strike into the net
Kane: Our best attacking performance
Harry Kane told BBC:
"It feels amazing to be honest. What a crazy game. First and foremost, we're playing against a tough and organised team.
"After that first hydration break, we upped the level. The keeper made some unbelievable saves. But keep going and the moments would come.
"Whoever it is, we have hero moments.
"You have to stay patient in these games. The last couple of games were very similar. The pressure is higher. But it's about being ourselves.
"From an attacking point of view, it's the best performance so far. We are in the part of the tournament where we have to grind wins out.
"I just told the boys to enjoy it. Sometimes as an England player, you don't celebrate how you should. The same as every other nation, we're through - enjoy it. Over the 90 minutes, we're tough to handle."
Kane matches Lineker feat - Opta stats
- Harry Kane is the first England player to score a brace in a World Cup knockout stage match since Gary Lineker against Cameroon in the 1990 quarter-finals.
- Kane has scored 10 goals across his 11 major tournament knockout stage appearances since Euro 2020. It's three more than any other European player in that period, with Kylian Mbappe next on seven.
- Kane is now on 72 goals for club and country this season.