Born in El Paso, Texas, Jacobs came home in a time of 9.8 seconds, edging out Fred Kerley of the United States by 0.04secs.
Canada’s Andre De Grasse collected a second successive bronze in the event, having also finished third at Rio 2016.
It is the first time in 17 years that there has been a new winner of the men’s Olympic 100m crown following Usain Bolt’s retirement from the sport.
The Jamaican won gold at the Games in 2008, 2012 and 2016. Justin Gatlin was the last winner before Bolt, in 2004.
Yohan Blake missed out on the final and, with no Bolt, there were no Jamaicans in the last eight for the first time since Sydney 2000.
Blake, 31, the joint second fastest man in the world, then insisted it would be his last Games.
“It’s definitely my last Olympics. You know track is not easy. I won’t be ungrateful. I’ve gained a lot,” he said.
Hughes stormed to win his semi-final in 9.98s, the first time he had run sub-10 seconds this year, before his final DQ.
He said the false start was caused by cramp setting in his left calf as he took the set position in his blocks.
“With the cramp, I moved,” he told the BBC. “I feel that if I was in the final I would have walked away with a medal, no doubt about it.
“It was out of my control. It hurts a lot.”
After Daryll Neita reached the women’s final on Saturday it was the first time in 37 years – the Los Angeles Games – Team GB had athletes reach both 100m finals.
Reece Prescod was disqualified from the first semi-final for a false start while CJ Ujah finished fifth in 10.11s to bow out.