Securing their third team win in the discipline with just two time penalties and zero knocked-off rails, Britain was ahead of the United States with 4 and France with 7 penalty points.
The victory added to Britain's team eventing gold earlier in the week.
"I think the message back home is that we brought back gold medals, I think we've done our country proud," Brash told reporters.
"These horses are looked after unbelievably well. They're our family, you know. They're amazing athletes. We want equestrianism to stay in the Olympic Games. It's so important."
Riders were asked to steer their mounts over 14 obstacles reminiscent of Paris landmarks such as the Eiffel Tower and Notre-Dame cathedral, some as high as 1.65 metres (5.41 feet), in just under 80 seconds.
Charles, on 15-year-old bay gelding Romeo 88, had one of the day's best performances, incurring no penalties.
The crowd erupted in cheers when French first-time Olympian Olivier Perreau - called in as a replacement after another rider's horse failed the veterinary check - also had a perfect ride on Dorai D'Aiguilly, briefly lifting his team to top spot.
French President Emmanuel Macron was among the 16,500 spectators watching the horses perform against the backdrop of the famous Versailles chateau.
The track was tough compared to other events, riders told reporters. "It's the maximum level you could possibly have in terms of technical challenges," Germany's Christian Kukuk said.
Germany came fifth, missing out on a medal for the second straight time in a discipline they once dominated.
On Thursday, Germany had delivered the best performance, but qualification scores were erased for Friday's finals.
Defending Olympic champions Sweden, with top-ranked rider Henrik von Eckermann and his horse King Edward, finished sixth.
Mexico, who harboured hopes of a medal, had to withdraw on Friday morning, citing veterinary reasons regarding one of their horses. The team had no fallback option after another of their mounts had already failed pre-competition vet checks.