Butkevych, who became the first British woman to win a World Championship wrestling medal in 2012, won her opening -60kg fight, defeating Romanian Georgiana Filip.
But a defeat to Turkey's Hafize Sahin put her into the repechage and she won through to beat Belarusian Nadzeya Mikhalkova inside 18 seconds to secure a podium place.
Butkevych, 28, won bronze at the event two years ago in Germany and will now look to this summer's Commonwealth Games, though her preferred weight category isn't on the schedule, so she will target the -58kg or -63kg divisions.
Shock incident sees India's Dahiya bitten on arm by rival during wrestling semi-final
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Shocking footage reveals full-jawed bite on arm in wrestling medal bout
Tokyo 2020 - ROC vs Uzbekistan - Wrestling Men's Freestyle 74kg - 1/4F - Olympic Highlights