Rassol returned with figures of 23-9-41-7 to trigger Assam's batting collapse as J&K shot the home side out for just 165 from 64.3 overs in their first innings in reply to the visitor's 323 all out at the Nehru Stadium.
J&K took a vital 158-run first innings lead. They ended the day at one for no loss in their second essay, stretching their overall lead to 159 with all the 10 second innings wickets intact.
For Assam, only captain and opener Dheeraj Jadhav put up a semblance of fight with a stoic 150-ball 69 before he became Rassol's second victim and third of his team.
Jadhav put on 55 runs with Pallavkumar Das (28) for the opening wicket before stitching another 60 runs for the third wicket with Sibsankar Roy (25).
But, once Jadhav was out, Assam folded their innings in quick time with seven wickets falling for the addition of just 39 runs with Rassol claiming five of them.
Earlier, resuming at overnight 240 for seven, Jammu and Kashmir were all out for 323 with Obaid Haroon (52) completing his half century.
He added 17 runs to yesterday's 35 while the other overnight batsman Waseem Raza added 28 runs today to contribute 40 runs to his team total.
For Assam, Arup Das took two more wickets today to end with innings figures of six wickets for 87.
Jammu and Kashmir: 323 all out in 109.1 overs (Obaid Haroon 52; Arup Das 6/87) and 1 for no loss in one over. Assam: 165 all out in 64.3 overs (Dheeraj Jadhav 69; Parvez Rassol 7/41).