The only saving grace for the hosts was half-centuries from wicket-keeper Thilak Naidu (51) and B Akhil (55) who put on 70 runs for the sixth wicket after the team, opting to bat first, slumped to 58 for five.
Salvi ended with impressive figures of 19.4-8-26-4, while Hazare's read 20-8-47-4.
Mumbai stumper Vinayak Sawanth also played a key role in Karnataka's downfall in the first innings taking four catches as the visitors asserted their supremacy and took the first day's honours.
Salvi removed openers Robin Uthappa (20) and Rowland Barrington (0). Hazare was also equally aggressive dismissing skipper Sujith Somasundar (2), Sudhindra Shinde (18) and Stuart Binny (0) leaving the local team in a crisis.
Mumbai were one without loss in the one over they faced, with S Kukreja batting on one and in-form Wasim Jaffer, who hit a double century in the first match against Delhi, yet to open his account.
Brief scores: Karnataka all out 186 in 86.4 overs (Akhil 55, Naidu 51; Salvi 4/26, Hazare 4/47, Nilesh Kulkarni 1/36, Ramesh Powar 1/59).