Farah, one of 12 contenders for the award, has been shortlisted five times before and enjoyed his previous highest finish of third in 2011.
After the show went off air, Farah spoke to those inside the arena.
Appearing close to tears, Farah said he was shocked to win because of the quality of the other athletes up for the award.
“It is pretty amazing and hard to think about,” he said.
“I didn’t imagine I was ever going to win this but anything can happen. If you work hard you can achieve your dreams.
“I am sorry I could not be there. My kid has been not well.
“I just cannot believe I have won.”
A third successive World Championships 10,000m gold medal was the highlight of a year in which Farah also won a world 5,000m silver, missing out on a fifth major championships distance double in a row.
The Somali-born Londoner received a knighthood from the Queen at Buckingham Palace in November.
He bowed out from his track career with a 5,000m victory at the Diamond League event in Zurich in August, and will now concentrate on road races.
Farah took the prize with 83,524 votes – 2,957 more than second-placed Rea, while Peacock took third with 73,429, just 18 more than boxer Anthony Joshua.
