Felix ran the second leg as the USA team - also featuring Wilbert London, Courtney Okolo and Michael Cherry - won in three minutes 9.04 seconds, a world record for the new event, to finish ahead of Jamaica and Bahrain.
The team draped themselves in American flags as they celebrated their win but, with almost no spectators in the Khalifa stadium, went straight down the tunnel instead of completing a lap of honour.
I am a clean athlete, says Hassan
Seven of the eight teams chose to run the first and fourth legs with men while Poland went for men in the first two legs and women in the last two.
That meant the Poles finished the second leg with a healthy lead which was cut back in the third leg and obliterated within the first 100 metres of the fourth as Justyna Swiety-Ersetic was overtaken by Cherry. Poland eventually finished fifth - just behind Great Britain's team of Rabah Yousif, Zoey Clark, Emily Diamond and Martyn Rooney.
Felix, 33, has also won three world titles in the women's 200 metres, one in the 400 metres, three in the women's 4x100 metres relay and four in the women's 4x400 metres relay.
Elsewhere, Russian athlete Anzhelika Sidorova, competing as a neutral because of her country's ban from athletics, soared to the women's world pole vault title with the best leap of the year on Sunday.
Sidorova pipped Sandi Morris by vaulting 4.95 metres with her third and final attempt, leaving the American with a silver medal for her third successive major championship after clearing 4.90.
In the men's 200m, Britain's sprinters Adam Gemili, Miguel Francis and Zharnel Hughes all advanced from their heats to qualify for Monday's semi-finals.
And in the men's triple jump, USA's Christian Taylor won a fourth ttitle with an effort of 17.92m, ahead of compatriot Will Claye with 17.74m Burundi's Hugues Fabrice Zango, who jumped an African record distance of 17.66m.
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