The 33-year-old, who fought back from a stumble to claim the 10,000m last week, said his victory in the 5,000m on Saturday night had been the most satisfying of the four golds.
Farah won his second straight 5,000m in 13min 3.30sec to match the feat of Finland's Lasse Viren who retained the same Olympic titles in 1976.
"Oh my God I can't believe it," said Farah. "It's every athlete's dream but I can't believe it. "My legs were tired after the 10,000m and people had to bring me food in my room.
Farah, who trains in Oregon under ex-marathon great Alberto Salazar, said watching Ethiopian distance legend Kenenisa Bekele medalling had been an inspiration.
"When Bekele won all those medals I said I just want one.
"If you have dreams they can come true and I always wanted to achieve these for my kids because for so much of the year you don't see them and thus you want to show them something or rather the reason for the absences."
- Never lose -
==============
Kenyan-born American Paul Chelimo took silver in a personal best of 13:03.90. He was initially disqualified but quickly reinstated.
Dejen Gebremeksel, the Ethiopian silver medallist behind Farah in London four years ago, and Gebrhiwet set out on a fast pace, Farah sat at the end of the strung-out field at the Olympic Stadium in perfect conditions.
Chelimo, one of three foreign-born Americans in the field along with the 41-year-old former Kenyan Bernard Lagat and Somali-born Hassan Mead, was happy tagging along in third as Farah moved up to sixth.
The Ethiopians alternated the lead, with Kenyan-born Bahraini Albert Rop also in the mix.
With five laps to run, Farah was well positioned in second and then took over the lead, immediately dropping the pace so the pack bunched back up.
Farah went through the bell head-to-head with Gebrewihet, who punched down the back straight.
But Farah held firm and surged again from 200m. Chelimo looked like he might have had the beating of the Somali-born Briton as they entered the home stretch. But Farah found an extra step in his stride, timing 52.83sec for the final lap for a famous win.
"When I hit the front I wasn't going to let them past me," said Farah. "I wasn't going to let the inside lane go. I didn't want to get boxed in.
The victory sealed a distance double for Farah last achieved when Viren triumphed in the 1972 and 1976 Games in Munich and Montreal.
In a stellar career, Farah has already achieved the world double-double at the 2015 and 2013 championships in Beijing and Moscow.
Salazar has faced doping suspicions, but no charges have ever been made Farah has stayed loyal to the Cuban-born coach.
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