Bahamas, Olympic champions on the same track five years ago, also failed to qualify for the final as did Jamaica.
Botswana, among the title favourites, were fifth but quickly making up ground on the leaders at the end of the third leg of the second semi-final when Nijel Amos tried to hand the baton to Karabo Sibanda.
Sibanda took the baton but it slipped out of his hand and, as he went to pick it up, he collided with Brazil's Anderson Henriques who was left sprawled on the track and had to be helped away.
Sibanda went on to complete the race and Botswana still finished sixth, ahead of Japan.
The U.S., aiming for a seventh world title in a row, ignored the trouble behind them as they stormed away to win the heat in 2:59.23 ahead of Trinidad and Tobago and Belgium.
Belgium snatched third spot from Britain when Kevin Borlee pipped Martin Rooney on the final straight.
Britain still went through as one of the fastest losers as did France who were fifth in the heat in 3:00.93 but still faster than Spain, winners of the first semi-final in 3:01.72.
Spain were followed by Poland and Cuba, the other qualifiers for Sunday's final. Jamaica were fourth while Bahamas, who also won bronze in Rio de Janeiro last year, trailed home in sixth.
(Writing by Brian Homewood; Editing by Toby Davis)
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