Gujarat eventually folded after 165.2 overs.
In reply, Hyderabad were one down for 50 in 12 overs at the draw of stumps.
While Patel's 111 came off 144 deliveries, Juneja negotiated 159 deliveries in his 106-run knock.
Resuming at the overnight score of 254 for two, which included opener Smit Patel's 114, Gujarat never seemed in trouble.
But Bhargav Merai fell short of what could have been the fourth century in Gujarat's scorecard, falling for 78 that came off 157 balls and included 13 hits to the fence.
Hanuma Vihari dismissed him but that did little to stop Gujarat's onslaught as Patel and Juneja combined for a 178-run fifth-wicket partnership.
The two batsmen frustrated the Hyderabad bowlers, who were ordinary at best. While Patel struck 14 boundaries, Juneja sent down 16 fours during his stay at the crease.
Eventually Patel became Vihari's second victim, while Juneja was sent back by Lalit Mohan.
After the duo's departure, Gujarat's tail did not wag much as Hyderabad stopped the hosts from crossing 600.
Replying to the target, Hyderabad were not off to the best of starts as Bhavesh Baria induced an edge from opener Kolla Sumanth to dismiss him for 18.
At close of play, skipper Akshath Reddy (30) and Mohammed Khader (2) were holding fort for Hyderabad.
Gujarat 1st Innings: 566
Hyderabad 1st Innings: 50/1 in 12 overs (A Reddy 30 batting).