Kallis claimed Guyana captain Ramnaresh Sarwan's (7) crucial wicket early in the innings before scalping two batsmen in the 16th over after Kumble elected to field after winning the toss.
Kallis (3/16) was ably supported by Kumble, Praveen Kumar and Nayan Joshi by chipping in with a wicket each as RCB bowlers tied down the Guyanese batsmen, most of whom lacking in international exposure, all through the innings.
Barnwell top-scored with a 35-ball 30 while the next highest scorer was opener Sewnarine Chattergoon with an 18.
Image: Royal Challengers Bangalore's Jacques Kallis, left, celebrates with teammates after dismissing Guyana's batsman Sewnarine Chattergoon for 18 runs during the Champions League Twenty20 cricket match at the Supersport Park in Pretoria
Scores | Champions League