Chebet won in 14 minutes and 28.56 seconds as the Netherlands' Sifan Hassan (14:30.61) initially claimed silver and Italian Nadia Battocletti took bronze (14:31.64).
Chebet produced a thrilling finish to win the first medal for her country at the Paris Olympics and smiled widely as she crossed the line after an exhausting performance.
World champion Kipyegon had finished second in 14:29.60 but was disqualified after a clash with Ethiopian world record holder Gudaf Tsegay on the penultimate lap.
Her silver medal, however, was later reinstated following an appeal by the Kenyan team, pushing Hassan down to the bronze-medal position and bumping Battocletti off the podium.
"Congratulations to Faith Kipyegon, her silver medal has been reinstated," Athletics Kenya said on social media platform X.
"Kenya's team officials led by Milka Chemos, Bernard Ouma and Team Manager Peter Angwenyi successfully appealed and won her case."
Chebet, runner-up at the 2022 World Championships in Eugene and bronze medallist in Budapest last year, took the early lead and hung in with Kipyegon through much of the race before surging past her off the final bend.
Tokyo champion Hassan was in fifth place heading into the final 400 and dug deep to try to hang on to her Olympic crown but could not match the Kenyans' speed around the final turn.
Chebet hugged and cheered with Kipyegon as the pair donned Kenyan flags to wild applause from the packed Stade de France.
It quickly became clear, however, that Kipyegon would not join her on the podium as her name disappeared from the results list and she was later marked disqualified.
"I am so sad for Faith and I hope justice will be done. Because Faith did not do anything bad," Chebet told reporters. "I think she will get it (the medal) in the end."
Hassan stood squinting at a monitor showing the results as Battocletti stood by in a confusing scene minutes after the race.
"I'm happy for you guys - for me, I don't care," Hassan said.
The Dutch woman will continue a gruelling Paris campaign with the 10,000m and marathon on her agenda.
"It's my crazy thing, I just want to complete it. I know people have done (three events) before but not in the 10,000 metres and marathon. I'm very curious," said Hassan.
"Could I podium? Could I even complete (the races)? I'm trying to fight with myself."
