Chebet -- nicknamed "the smiling assassin" -- bided her time before accelerating past compatriot Faith Kipyegon down the home straight to take gold in 14min 28.56sec at the Stade de France.
Kipyegon, the reigning 5,000m world champion, crossed the finish line in second place but was swiftly disqualified for a shove in the closing laps.
That ruling saw Dutchwoman Hassan moved into the silver medal position with a time of 14:30.61 while Italy's Nadia Battocletti took bronze in 14:31.64.
Hassan, 31, had been aiming to become the first woman in history to win 5,000m, 10,000m and marathon gold at the same Olympics, emulating the feat achieved by men's distance running legend Emile Zatopek at the 1952 Helsinki Games.
But after deploying her favoured tactic of staying at the back of the field for most of the race, Hassan was unable to live with the pace of Kenya's Chebet and Kipyegon in the final stages of a thrilling race.
Kipyegon edged just clear of Chebet coming down the home straight but was unable to respond when her compatriot found an extra gear and swept past her.
Hassan crossed the line in third place but was elevated to silver moments after the finish following Kipyegon's disqualification.
Chebet's victory was the latest milestone in a dazzling season for the 24-year-old Kenyan, who set a world record in the 10,000m in Eugene, Oregon in May.
Chebet will now look to seize Hassan's Olympic 10,000m crown when the race takes place on Friday.
