An occasion that started with boos for the heavily-criticised manager on Saturday (Sunday AEST) ended in a breathless bronze medal, with the wounds from their semi-final collapse to Argentina still raw.
Saka's brace followed goals from Declan Rice and Ezri Konsa in a remarkable first half that was followed by a scarcely believable second period in the Miami heat and humidity.
France came out swinging in Didier Deschamps' final match in charge as Kylian Mbappe scored a brace, taking charge of the Golden Boot race and becoming the top scorer in World Cup history with 22 goals.
Substitute Bradley Barcola scored the other and Les Bleus continued to pepper England's goal as Tuchel stared down the barrel of even greater humiliation than three days ago.
But Saka scored a late penalty to complete his treble and substitute Jude Bellingham kept his cool after Ousmane Dembele's earlier added-time strike.
Harry Kane and Bellingham were named on the bench as Tuchel made seven alterations, including surprisingly starting Rice at the end of a summer that has seen him manage a hamstring injury.
But after just three minutes the move paid off as he burst forward after cutting out a pass before placing an exceptional shot past Mike Maignan from 20 yards.
Rice swung over a corner in the 18th minute and Konsa rose to head home a second.
Maignan prevented Marcus Rashford scoring a magical long-range effort and then denied him on the break, only for the forward to help keep the move alive and cut the ball back for Saka to sweep home.
England added a fourth in added time when Eberechi Eze's exceptional pass was met by a similarly impressive Saka finish.
Deschamps made four halftime changes and Mbappe pulled a goal back inside three minutes after Michael Olise slipped him in to score a fine first-time finish.
Barcola was among those brought on and he added another in the 54th minute, having been brilliantly slipped in by Mbappe.
Olise, Adrien Rabiot and Dembele all threatened as England faltered and Dayot Upamecano saw an effort saved before a world-class move ended with Mbappe scoring in the 66th minute after collecting a return pass from Olise.
England had no control whatsoever and Olise twice uncharacteristically screwed wide as France sought an unthinkable leveller.
Substitute Bellingham came close to seemingly settling proceedings shortly after his introduction, with England then getting a chance from the spot when Malo Gusto fouled Djed Spence.
Saka held his nerve to score his penalty and complete his hat-trick in the 87th minute, but Dembele brought France back to within a goal before England's star man Bellingham settled things with his seventh, and most stunning, goal of the tournament.
It provided a wild end to a summer that will forever have the nagging feeling of what might have been.
