The long-striding Clement led down the home straight and was still clear as he entered the closing metres. But he had to lean into the tape to deny Kenya's Boniface Tumuti, charging up fast two lanes to his right.
"I fought the last 100 metres because I knew the guys would be coming. The last few metres I just dug down deeper," Clement said after clocking 47.73 seconds to win by 0.05 of a second.
Culson, 32, bronze medallist in London in 2012, blew his chance by jumping the gun, and briefly sat sobbing by the side of the track before trudging away.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"When he false-started I just said to myself: 'Refocus'," Clement told reporters. "The adrenaline rush just zapped him out.. I tried not to let it affect me and just focus on the 10 hurdles in my lane again."
Tumuti said he had "lost the gold dream" with a miscalculation on the approach to the seventh hurdle.
Turkey's Yasmani Copello won the bronze and told reporters: "I think my mother is crying. I am the younger son and I'm making her dreams come true. She will be crying a lot."
It was Clement's first major championship medal since he won the 2009 world title, having taken the silver at the Beijing Olympics the year before.
In London four years ago he came eighth, after suffering hernia problems and undergoing surgery twice.
"Coming out here in 2016 for me is a redemption year. I'm just really honoured to get the gold medal," the 30-year-old said, describing himself as still young for a hurdler and intent on competing in Tokyo in four years' time.
Thomas Barr of Ireland came fourth, narrowly failing to win Ireland's first Olympic medal on the track since Sonia O'Sullivan took silver in the 5,000m at the Sydney Olympics in 2000.
"If you can't be happy with a lifetime best ... I would be lying if I said I was complaining, but I was so close - 0.5 of a second, and I think I could have found it," the Irishman said. "Fourth is the worst place, as you are so close."
(Reporting by Mark Trevelyan; Editing by Neville Dalton, Alison Williams and Nina Chestney)
