The veteran Indian leg spinner dismissed Derbyshire's middle order bat Michael James Di Venuto for 34 as Surrey won their second successive Pro40 game.
Kumble's figures read 8-0-35-1.
Earlier, Derbyshire won the toss and opted to bat first in the match and some grity batting by the middle-order -- especially Chris Taylor's 44 -- took the visitors to 207 after they suffered early jolts. Derbyshire were all out in 39.4 overs and Nayan Doshi proved to be their chief nemesis as he took a five-wicket haul conceding 30 runs in his eight overs.
Playing at home, Surrey had little trouble in reaching the target courtesy a superb 68-ball 106 by Ali Brown and equally well struck 66 by James Benning. The hosts reached the target with 12.4 overs to spare