Electing to bat, J&K lost three of their top order batsmen cheaply to be struggling at 20 for three.
But one-down batsman Ian Dev Singh and Rasool steadied the boat as the duo put on a solid 123-run stand for the fourth wicket before Puneet Bisht contributed with a crucial 40 to take the team past 200-run mark.
In reply to J&K's modest first innings total, Goa failed to get off to a good start and kept losing wickets at regular interval to be reeling at 43 for five in 17 overs.
At the close of play, Amit Yadav (0) was at the crease.
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Ram Dayal (2/10) and Rasool (2/18) took two wickets each.
Brief scores:
J&K 1st innings: 227 all out in 70.3 overs (Parvez Rasool 82, Ian Dev Singh 67; Shadab Jakati 4/48, Amit Yadav 3/43).
Goa 1st innings: 43 for five in 17 overs (Ram Dayal 2/10, Praveez Rasool 2/18).