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.
"It is the first medal in 60 years, the best World Cup on foreign soil so I hope the players can be proud of that in some time," German Tuchel said.
"We are fiercely competitive, so we almost don't allow ourselves to be proud with third place.
"Eighteen months ago we set the highest goal, the highest dream to chase. We were very, very ambitious with our dream to make it to the final and win the World Cup so it is very, very painful if you miss out."
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.
"We were not expecting such a first half," Deschamps said through an interpreter.
"We were deeply disappointed, gutted after the semis. I don't want to sum up my career on this moment."
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 and placed 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 and the Real Madrid striker scored again in the 66th after collecting a return pass from Olise.
England then got 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 substitute Bellingham settled things with his seventh, and most stunning, goal of the tournament.
"In the end, we didn't win, and that's a shame for the coach," Mbappe said.
"The first half gives the impression that we let him down - that's not at all how we wanted him to feel. This match isn't going to tarnish Didier Deschamps' legacy."
With the AP.
