Joshi took six wickets for 73 runs and Kumble scalped three for 89 as the visitors, resuming from their relatively strong overnight position of 213 for two, slumped to 258 all out replying to Karnataka's 283.
At stumps, the hosts reached 196 for five wickets off 75 overs in their second innings, ahead by a total of 221 runs, with a day's play remaining.
Karnataka skipper Sujith Somasundar scored 52.
Wicket-keeper V S Thilak Naidu was batting on 44 and B Akhil on 14 at the close.
Earlier, Joshi triggered a batting collapse that saw wickets falling like nine-pins. Delhi skipper Virender Sehwag, batting on 40, added only 16 runs before being bowled by Joshi, who turned out to be wrecker-in-chief.
Kumble then joined the party removing Mithun Manhas (13) and Varun Kumar (zero) to guide his team back into the reckoning.
Brief scores: Karnataka 283 and 196 for five in 75 overs (Somasundar 52, Robin Uthappa 20, Bharath Chipli 20, Naidu batting 44; Chaitanya Nanda 3/61, Sehwag 2/42) vs Delhi 258 all out (Akash Chopra 84, Shikhar Dhawan 82, Sehwag 56; Joshi 6/73, Kumble 3/89).