Asked to bat, the visitors posted 246 for five before the day's proceeding was called off early due to rains with 21 overs still remaining to be completed.
Electing to bowl, Goa got an early break-through when medium pacer Rituraj Singh dismissed opener Tanmay Agarwal (14) with the visitor's score at 25.
But skipper Hanuma Vihari (82) and other opener Akshath Reddy (61) then steadied Hyderabad innings with a 113-run second wicket partnership.
The partnership was eventually broken by spinner Shadab Jakati, who caught Reddy of his own bowling.
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Vihari, in the meantime, completed his half century but became Rituraj's second victim of the day just 18 runs short of his century.
Hyderabad also lost two quick wickets in the form of Bavanaka Sandeep (22) and A Ashish Reddy (8).
Hyderabad were dealt another setback when wicketkeeper Habeeb Ahmed got injured and had to retire hurt on 5, before bad light and rains drew premature curtains to the day's proceedings.
For Goa, Rituraj (2/52) clinched two wickets while Jakati and Amit Yadav picked up a wicket each.
Brief Scores:
Hyderabad 1st innings: 246 for five in 69 overs (Hanuma Vihari 82, Akshath Reddy 61; Rituraj Singh 2/52).