The 33-year-old overcame a mid-race fall and powered clear of the field in the final 100m to win in 27 minutes five seconds.
Kenya's Paul Tanui took silver, with Ethiopia's Tamirat Tola in third.
Farah returns to the track on Wednesday as he begins the defence of the 5,000m title he also won at London 2012.
"I've won an Olympic gold for three of my children," he said. "Now I'd like to win the 5,000m gold for my little boy."
Should he win that race, Farah will become the first man since Finland's Lasse Viren in 1976 to retain two Olympic distance titles.
He has already achieved the world double-double, having successfully defended his 10,000m and 5,000m titles in Beijing last year.
Farah looked comfortable at the back of the field in Rio, choosing his moment to pick his way through the pack.
His hopes briefly looked in danger when he was accidentally tripped by training partner Galen Rupp with 16 laps to go, but he recovered to surge past Tanui on the home straight.
"I wasn't going to let it go," he said. "I got up quickly. I thought about my family. It made me emotional. I thought 'get through, get through'. I believed in myself."
Farah's compatriots Andy Vernon and Ross Millington finished 25th and 31st respectively.
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