The 25-year-old - the youngest winner of the London Marathon - was congratulated by Radcliffe at the finish line as her sensational year reached new heights.
Ababel Yeshaneh came second with a personal best of 2:20:51, almost seven minutes behind Kosgei, with fellow Ethiopian Gelete Burka completing the top three.
"I'm feeling good, happy, I was not expecting to run like this," Kosgei said. "The course was good - a little bit of wind, which pushed us a bit, but it was okay."
Kosgei's incredible run completed a supreme double for Kenya, with Lawrence Cherono winning the men's race.
Ethiopia's Dejene Debela looked well placed to clinch victory when he made a break for the line in the last kilometre, but he went too early.
Cherono took full advantage, driving clear in the final stretch to cross the line with the clock on 2:05:45. Asefa Mengstu came third.
However, it was a frustrating day for 2018 champion Mo Farah, who recorded his worst marathon run in six races at the culmination of a difficult week.
Farah - who accused the media of having a "clear agenda" against him when he was questioned over the conduct of his former coach Alberto Salazar, who has been banned for four years after being found guilty of doping violations - completed the race in eighth with a time of 2:09:58.