Greg Rutherford said he was "gutted" not to have defended his Olympic long jump title as the Briton finished third behind winner Jeff Henderson,
The American, 27, won gold by recording 8.38m with his final attempt.
South Africa's Luvo Manyonga won silver with 8.37m, while Rutherford took bronze with a final leap of 8.29m.
"I came here to win but couldn't quite get it together," said an emotional Rutherford, 29. "I'm very disappointed. Bronze is not good enough for me."
Rutherford was nearly edged out of third by Jarrion Lawson, but the American's hand brushed the sand on what would have been a medal-winning final jump.
Rutherford's performance, allied to Jessica Ennis-Hill's silver in the heptathlon, meant Britain could not repeat London 2012's Super Saturday, although Mo Farah did retain his 10,000m title.
Elsewhere, Britain's Matthew Hudson-Smith booked his place in the 400m final with a perfectly timed run from lane eight, finishing strongly down the home straight to clock 44.48 and a new personal best.
But neither fellow Briton Michael Rimmer nor Ireland's Mark English could make the 800m final, while Britain's Luke Cutts went out in pole vault qualifying.
Chief sports writer Tom Fordyce in Rio:
"Rutherford might not have retained his Olympic title, but in one of the great long jump finals - in terms of drama if not distance - he proved his immense championship character with that last-round leap.
"He was just two centimetres down on the jump that won his gold in London and just 10cm from another gold, having had his preparation badly disrupted by the whiplash injury he suffered at the Birmingham Diamond League."