Tamgho's 18.04 metre leap in the last round put him behind only American Kenny Harrison (18.09) and Jonathan Edwards's 1995 world record of 18.29 and the Briton's earlier 18.16 in the same Gothenburg event on the all-time list.
But for a red flag earlier in the evening the 24-year-old Frenchman could well have been on top of the pile.
It was an extraordinary finale to a gripping competition that began with 20-year-old world leader Pedro Pablo Pichardo setting the early pace with 17.68m, only for Tamgho to match it exactly and move ahead on countback.
Tamgho, the indoor world record holder with 17.92m, then landed two fouls around the 18-metre mark, one of them a monster, before getting everything right with his last.
"I don't know what to say," said Tamgho, who missed last year's Olympics with injury and in 2011 was banned for six month by his federation after an altercation with a female team mate.
"A big jump was really needed to take the win.
"Jonathan Edwards congratulated me and said that one of my fouls was above his world record."
Cuban Pichardo took silver with 17.68 and Will Claye of the United States bronze with 17.52, well clear of out-of-sorts compatriot and world and Olympic champion Christian Taylor in fourth.
(Editing by Ed Osmond)
