Farah, 34, is stepping up to the marathon and had hoped to add to the doubles he won at the 2012 and 2016 Olympics, and the 2013 and 2015 world championships.
Kenyan-born Chelimo took the bronze in 13:33.30 while Farah's late burst also consigned another Ethiopian Yomif Kejelcha to fourth place in 13:33.51.
"I gave 110 per cent", he said in the immediate aftermath.
Edris finished in 13mins 32.79secs, with the United States' Paul Chelimo third. It's not an excuse, but it took a lot more out of me than I realised.
Claiming to have been exhausted after his 10,000m efforts and 5,000m heats, he added: "Tactically, I was trying to cover every move".
Quite simply, Farah will go down in history as Britain's best-ever athlete and one of the greatest the world has ever witnessed. The better man won.
With nine laps to go Farah's fellow Brit Andrew Butchart briefly led with the pace slow as Edris and Cyrus Rutto made the moves to go up with Farah.
Roared on again by a home crowd, the Briton had looked comfortable from the start.
As the group went through the 3,000m mark Australian Patrick Tiernan was the one who chose to change the pace and he opened up a little gap between Chelimo, Farah and the rest of the pack. 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.
But when he starts his career as a marathon runner, Farah will be able to reflect on having created an wonderful legacy with his four Olympic and six World Championship gold medals. That he finishes with silver on the track clearly wasn't the farewell he had planned.