At stumps, the hosts were 242 runs ahead of Kerala with five wickets remaining.
Hyderabad rode on Vihari's 296-ball 145 and Quadri's patient 131-ball 74 to ensure they did not collapse in their second essay.
After the loss of Akshath Reddy for nought, Vihari stabilised the innings with Bavanaka Sandeep to raise a 78-run partnership for the second wicket.
Medium pacer Nizar Niyas broke the stand when he had Sandeep caught by Prasanth for 30.
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Vihari then found an able ally in Quadri and both mixed caution with aggression to put on a century-plus stand for the fourth wicket and guide their team to safety.
Quadri was the first to depart when his timbers were disturbed by Prasanth. Quadri's innings was laced with 10 hits to the fence.
Vihari remained focused on the task at hand and stitched another valuable 76-run partnership with Amol Shinde for the next wicket to take the team's total at 300 for four.
Vihari was unlucky to have missed out on his 150 when Niyas returned to scalp his prize wicket.
Shinde (48 batting) and Aashish Reddy (19 batting) were at the crease when stumps were drawn for the day.
For Kerala, Niyas was the most successful bowler with figures of three for 82.
Brief scores: Hyderabad: 201 and 329/5 in 106 overs (Hanuma Vihari 145, Syed Quadri 74; Nizar Niyas 3/82)
Kerala: 264 all out in 83.4 overs (P Prasanth 106; Ravi Kiran 4/81).