Germany's Michael Jung had one fence down on La Biosthetique Sam, which saw him drop to bronze.
Jung's fault meant Cook had two fences in hand, although she worried British supporters with one down and four time faults.
But her score of 46.6 was enough to beat French's 47.3 to take the gold.
William Fox-Pitt was agonisingly close to adding a bronze to his team gold, but finished out of the medals in fourth on Idalgo.
Pathfinder and team debutante Nicola Wilson rode another solid round on Opposition Buzz but was disappointed with two fences down, finishing ninth on 64.7
"I have to blame myself for the second fence down," she said. "He's jumped well all week. It's been a fantastic experience to be part of the team."
Britain's other individual competitor, Lucy Wiegersma, also had two down and time faults for 70.2 to finish 13th on Woodfalls Inigo Jones.
"He's a young horse and wouldn't have seen an atmospheric stadium like this," said Wiegersma. "The crowd were very noisy but I'm sure he'll learn from it."
Team-mate Oliver Townend was forced to withdraw from the show-jumping phase when his horse, Flint Curtis, failed the vets' inspection.
Cook overwhelmed to win gold
The 26-year-old Badminton and Burghley winner, who had had been the joint leader after the dressage, still received his team gold as his elimination meant his score did not count.
Britain's hopes for the team gold medal were significantly boosted when France's Jean Teulere did not present his horse Espoir de la Mare for inspection on Sunday morning.
Defending European individual champion Nicolas Touzaint was eliminated during Saturday's cross-country event, which meant France could not finish the championship with the mandatory three riders and were penalised 1000 points.
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