Kumble grabbed the last two wickets of the home side, who were all out for 165 in less then 50 overs after Surrey ended with 576 in their first innings. With this win, Surrey have further consolidated their position at the top of Division Two with a 34-point lead over their rivals in the match.
For Worcestershire, Zaheer Khan took his season's tally to 66 wickets after scalping three in the match.
Surrey, who were cruising ahead with a 192-run lead at the loss of just three wickets, suffered a middle-order collapse and added just 80 runs to their overnight total before being all out.
Rikki Clarke got past 1,000 runs this season with a quickfire 40 but found little support from the other batsmen as Surrey were dismissed for 576.
In reply, Worcestershire lost both their openers within two overs. Vikram Solanki (56) and Gareth Batty (37) were the lone fighters for the hosts who were bundled out for 165. The loss also ended Worcestershire's six-match no-loss streak in which they won five and drew one game.