Kulkarni returned his third five wicket haul of the season, taking 5 for 39 with his left-arm slows, while off break bowler Powar grabbed 4 for 70. The duo helped Mumbaitake a handy 126-run first innings lead on a spinner's track.
Karnataka, who managed to send back the last two Mumbai batsmen after the hosts added only 19 runs to their overnight first innings score of 273 for 8, were shot out 20 minutesbefore close in 83.2 overs.
They lost their last eight wickets for 50 runs in a dramatic post-lunch collapse.
Karnataka, sailing smoothly at 82 for no loss and then 116 for two replying to Mumbai's modest first innings total of 292, were unable to decide between attack and defence afteradjourning for lunch at 77 for no loss.
The visitors were given a sound start by openers Barrington Rowland (43, 6 fours and a six) and Robin Uthappa (39 with 4 fours and a six) and then lost the duo in the spaceof four balls.
They never fully recovered from the double strike and lost wickets in clusters of two and three thereafter.
Before stumps, they fought back a bit by dismissing Wasim Jaffer in Mumbai's second innings score of 2 for 1.
Scores: Mumbai 292 (V Mane 69, V Indulkar 47, N Shetty 65; S Joshi 6 for 102) and 2 for 1. Karnataka 166 (B Rowland 43, R Uthappa 39; R Powar 4 for 70, N Kulkarni 5 for 39).