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 took the early lead and stayed grouped up with compatriots Kipyegon and Margaret Kipkemboi through much of the race, keeping her cool as Kipyegon and Ethiopian Gudaf Tsegay collided at the start of the penultimate lap.
Tokyo champion Hassan was in fifth place heading into the final 400 metres and dug deep to try to hang onto her Olympic crown but could not match the Kenyans' speed around the final turn.
Chebet hugged and cheered Kipyegon as the pair donned Kenyan flags to wild cheers from the packed house at the 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, barring an appeal.
