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Having reached the semi-finals in 2018 and finished as runners-up at last year's European Championship, Gareth Southgate's men were dreaming of a mouthwatering semi against Morocco.
England's hopes were dashed by Aurelien Tchouameni's early thunderbolt, only for Kane to level from the spot and equal Wayne Rooney's goalscoring record for the national team.
France's all-time scorer Olivier Giroud restored the World Cup holders' lead and it proved to be the winner after Kane blazed over a late penalty in a crushing 2-1 exit.
England have a long history of costly spot-kick misses but normally those issues arise in shootouts, like the agonising Euro 2020 final defeat to Italy last summer.
Southgate is contracted until the next European Championship and this may well prove his final World Cup match in charge - a night when his players rallied impressively after falling behind for the first time in Qatar.
Both managers named unchanged sides in Al Khor and it was a cagey start to a match that came to life with Tchouameni's thunderous drive from distance.
Penalty appeals for a Dayot Upamecano foul on Kane were also ignored after the VAR concluded the decision occurred outside the box.
But England deservedly drew level when Tchouameni upended Saka early in the second half, with Kane beating Hugo Lloris to move level with Rooney's all-time record of 53 goals.
Giroud put Didier Deschamps' side back ahead, only for Theo Hernandez's clumsy challenge on substitute Mason Mount to bring another penalty after referee Wilton Sampaio reviewed the pitch-side monitor.
But Kane found the Al Bayt Stadium stands rather than the net, meaning England are packing for home rather than preparing for a third successive semi-final in major competitions.
Southgate's side responded impressively to Tchouameni's early gut punch with Lloris smothering a Kane effort and then tipping Jude Bellingham's stunning half-volley over early in the second half.
In the 54th minute, Saka was sent tumbling by a Tchouameni challenge, leading referee Sampaio to point straight to the spot. A long delay followed but Kane kept his cool to send Tottenham teammate Lloris the wrong way.
Pickford prevented Adrien Rabiot from immediately putting France back ahead before Lloris saved from Saka and Kane, and Harry Maguire met a free-kick with a header that skimmed the outside of the post.
England had keeper Jordan Pickford to thank for keeping scores level when Giroud met an Ousmane Dembele header back across goal with a volley.
The veteran would not be denied for long as Griezmann swung a stunning cross over for Giroud to head home from close range via a slight touch off of Maguire.
Having inexplicably waved play on after Hernandez barged over Mount, the VAR informed referee Sampaio to check the monitor and the Brazilian pointed to the spot after reviewing the footage.
Kane would have broken Rooney's record with the goal and injected renewed hope, but was instead left distraught after blasting his 84th-minute spot-kick over the bar.
It was a gut punch that England could not respond to, on a night that ended with substitute Marcus Rashford seeing a free-kick 10 minutes into added time land on the roof of the net.
