Kane will bounce back 'stronger' from penalty miss - Henderson
Kane had equalled Wayne Rooney's all-time record of 53 goals for England by levelling the quarter-final in Qatar at 1-1 with his first penalty of the night.
But after Olivier Giroud put France back in front, the Tottenham striker blazed over when handed a second chance up against his club teammate Hugo Lloris.
"We know how many penalties Harry has scored for us, he scored the first one, how many goals he's contributed to even get us here," Henderson told ITV.
"He'll be stronger for it in the long run I'm sure.
"He's a world class striker, our captain and we wouldn't be here without him."
The defeat extends England's drought without a major tournament victory which stretches back to 1966.
France will face Morocco in the semi-finals and with Brazil, Germany, Spain and Portugal already eliminated, Henderson conceded a huge opportunity at glory had slipped through England's grasp.
"We felt really good, performances were really good. The focus and hunger was there," added the Liverpool midfielder.
"Unfortunately it wasn't our night. Give credit to France, who are a good team. I still felt it was there for us to win tonight."
Southgate said his players could have done little more to end France's reign as world champions in a tight contest.
"I think the performance (deserved better)," said Southgate. "In the end goals are decisive.
"I just said to the players I don't think they could have given any more, they've played really well against a top team. It's fine margins and things at both ends that have decided the game.
"I think the way they have progressed throughout this tournament has been fantastic.
Aurelien Tchouameni's stunning long-range strike fired France into an early lead, but Southgate's men believed they should have had a spot-kick before half-time when Kane was tripped by Dayot Upamecano.
France did not escape a second time when Tchouameni brought down Bukayo Saka and Kane confidently dispatched the penalty.
But just as England looked the more likely to go on and win the game, Didier Deschamps' men grabbed the decisive goal when Giroud's header deflected in off Harry Maguire.
England were handed a lifeline moments later when Brazilian referee Wilton Sampaio pointed to the spot once more after a VAR review.
But Kane wasted the chance to make history as his country's leading scorer and send the game to extra time.
"We win and lose as a team," added Southgate. "He's been incredible for us and so reliable in those sorts of situations.
"We wouldn't be here but for the number of goals he's scored for us."