Kumble conceded only 24 runs in his seven overs after Surrey -- having elected to bat first -- piled up 275 for seven. The veteran leg spinner also struck a four off the only delivery he faced.
Surrey captain Mark Butcher top scored for the hosts after openers James Benning (70) and Ali Brown (43) laid the foundation for the imposing total of 275.
Chasing the huge target, Leicestershire were also off to a good start as Darren Maddy (152) and Darren Robinson (32) put on 108 for the first wicket.
Maddy's 152 came in just 110 balls and included 18 fours and five sixes. However, his superb effort went in vain as Kumble and Nayan Doshi (4/63) combined break the visitors' batting line-up.
Kumble made the crucial breakthrough by dismissing Robinson and halting his threatening opening stand with Maddy.
Leicestershire could not recover from the middle-order collapse and were all out 33 runs short of the target in 36 overs.