Goa began the day at comfortable 281 for four and piled on the misery of the hosts with Asnodkar and overnight batsman Ravikant Shukla (87) batting superbly.
In reply to the huge total, Kerala reached 47 for one having lost opener VA Jagadeesh (19).
Sher Yadav had Jagadeesh caught behind.
Abhishek Hegde (26) and (2) P Anthaf (0) will begin Kerala's innings tomorrow.
Brief Scores:
Goa 1st innings: 512 for 8 declared. (S Kamat 107, R Asnodkar 100; C Shahid 4/149).
Kerala 1st innings: 47 for 1 in 20 overs.(A Hegde 26 batting; S Yadav 1/6).