Starting the day at 48 for no loss, Asnodkar in company of Amogh Desai (85), raised 178 runs before J&K got their first breakthrough.
Desai was the first one to return to the hut, falling prey to off-spinner Parvez Rasool.
His loss, however, did not have much bearing on the hosts as Manvinder Bisla put the visiting bowlers to test, cracking seven hits to the fence and one over it during his unbeaten stay at the crease.
Asnodkar, meanwhile, was more subdued in his approach scoring 103 off 279 balls with the help of 10 boundaries.
The 28-year-old, finally fell to leg spinner Bandeep Singh, but not before adding 58 runs for the second wicket.
At the close of the day, Bisla was not out 74 while Sagun Kamat was giving him company at 13.
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Brief scores:
Jammu and Kashmir 1st innings: 338 all out
Goa 1st innings: 288 for two in 119 overs (Swapnil Asnodkar 103, Amogh Desai 85; Bandeep Singh 43/1).