The Briton had won over 5,000m and 10,000m at the past two World Championships and Olympic Games, but Ethiopia's Edris denied him gold with a time of 13 minutes and 32.79 seconds.
Farah's last defeat in a global championships was the 10,000m at the worlds in Daegu, Korea, on August 28th six years ago.
Farah normally takes charge on the final lap but this time, following a tactical affair, he found himself playing catch-up over the final 400m as Edris and his fellow Ethopian Yomif Kejelcha led the way.
America's Paul Kipkemoi Chelimo finished in 13 mins 33.20 secs to take bronze. "I think I have gained a lot of experience after this race", he added. Perhaps this was a step too far into greatness, even for an athlete like Farah.
"There's more need for us to demonstrate that we have some incredible talent, but the athletes have also got to realise that they've also got a sizeable part to play in this".
"I gave it my all, I had nothing left", he said. They worked as a team.
This race was not as bruising and Farah looked in control until the final 100metres.
Last Friday night, he produced another bold if not ballsy display in the 10,000m, riding each wave of surge and counter surge with utter fearlessness: he very almost fell over himself too with 300m to run - yet held his nerve and ultimately devastating kick to win in 26:49.53, the fastest time in the world this year.
Johnson-Thompson tried to tap into the wave of emotion for Farah making her final attempt after the 5,000 race had started, but despite her efforts could not clear the bar.
Earlier, Sally Pearson produced one of the great comebacks in World Championship history, winning the 100m hurdles five years after the Australian women won Olympic title on this track, and after two years of crippling injuries which threatened her career.
Farah will run in Birmingham next Sunday before ending his track career in Zurich.