It had looked as though Spain were going to run away with victory at the Parc des Princes in Paris when they recovered from conceding an early goal to lead 3-1 by half-time.
Enzo Millot put France in front, but a Fermin Lopez brace and an Alex Baena free-kick turned the final completely on its head.
However, France staged a memorable comeback with Maghnes Akliouche pulling a goal back before Jean-Philippe Mateta converted a stoppage-time penalty awarded following a VAR review.
That took the final into extra time, where Rayo Vallecano forward Camello emerged as the hero by giving Spain the lead again in the 100th minute of an absorbing game, and then running away to seal their victory at the death.
Silver medallists in Tokyo three years ago, it is the second time Spain have won Olympic men's football gold after their victory in Barcelona in 1992 with a squad featuring Pep Guardiola and Luis Enrique.
France had hoped for a second gold of their own, 40 years after winning in Los Angeles, but instead have to settle for silver.
The victory completes a glorious summer on the pitch for Spain's men after their triumph at the Euros in Germany last month. They also recently won the Under-19 Euros by getting the better of France in the final.
However, Henry's team can leave with their heads held high at the end of a classic match, the scenario of which recalled memories of the 2022 World Cup final which France lost on penalties to Argentina after a similar comeback attempt.
Lopez brace
The final drama began when France went ahead after just 11 minutes thanks to Millot, who collected a scuffed Baena clearance just inside the box wide on the right and tried a first-time shot that caught out Spain goalkeeper Arnau Tenas.
However, Spain came from behind to beat Morocco in the semi-finals and they were level here in the 18th minute thanks to Lopez.
They strung together 26 passes in a move which culminated in Baena playing a ball into the middle for Lopez to finish.
Spain went ahead on 25 minutes when a Juan Miranda cross was not dealt with by the French defence and Lopez converted the rebound after Guillaume Restes had saved the initial effort from Abel Ruiz.
It was a sixth goal at the Olympics for Barcelona midfielder Lopez, who has had an outstanding tournament.
Penalty drama
The game had barely restarted when Spain won a free-kick just outside the area for a foul on Miranda, and Villarreal midfielder Baena sent the dead ball over the wall and into the net.
France had to chase the game but Mateta was denied by a superb Tenas save in first-half stoppage time, and Manu Kone headed against the bar just before the hour.
Tenas, of Paris Saint-Germain, distinguished himself again by tipping a Kone shot around the post, before France eventually scored a second goal on 79 minutes.
Michael Olise delivered a free-kick from the right and Akliouche's touch took the ball beyond the goalkeeper.
That set the scene for what felt like inevitable last-minute penalty drama, with the referee initially giving nothing when Benat Turrientes wrestled Arnaud Kalimuendo to the ground.
He changed his mind after a VAR review, and Mateta converted the spot-kick to equalise in the 93rd minute.
There was still time for Turrientes to hit the bar before the game moved into extra time, in which Spain went back in front, this time for good.
Adrian Bernabe released Camello and he beat Restes with a clipped finish to make it 4-3, before later running clear to score his team's fifth.
