Goa were 245 for nine in their second innings at stumps with an overall lead of 260 runs. Amit Yadav was unbeaten on 40 off 67 balls while Gauresh Gawas was yet to open his account.
Keenan Vaz and skipper Ravikant Shukla were the notable contributors for the visitors as Rasool spun a web around Goa batsmen to give his team a chance of an outright victory.
Local boy Rasool emerged as star of the day by taking four for 65 in 21 overs. The 24-year-old off-spinner always seemed to be on the lookout for wickets.
Amogh Sunil Desai became Rasool's first victim when the opener was trapped leg before for 25. Rasool then came for his second spell and broke the 75-run fourth-wicket partnership between Vaz and Shukla to leave the visitors struggling at 163 for five.
Few overs later, Rasool packed off Shadab Jakati for 20 as Goa were left tottering at 167 for seven.
Earlier, J&K, resuming at their overnight score of 282 for eight, could add only 20 runs to their first innings score before being bowled out for 302.
The hosts will now aim for an outright win or else they will have to be satisfied with a solitary point, having conceded the first innings lead.
Brief Scores:
Goa 1st innings: 317 all out in 97.1 overs.
Jammu and Kashmir 1st innings: 302 all out in 94 overs)
Goa 2nd innings: 245 for 9 in 73 overs (Keenan Vaz 59, Ravikant Shukla 46; Parveez Rasool 4/65).
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