Prem scored 170 off 364 balls while left-arm medium pacer Manukrishnan picked four for 20 as Kerala, after posting 400 in their first innings, reduced the hosts to 55 for four at the Maharaja Bir Bikram College Stadium.
Resuming on 275 for five, Kerala rode on Prem's knock, which contained 20 fours, to make a big total.
Tripura trailed the hosts by another 345 runs with six wickets remaining.
Brief scores:
Kerala 1st innings: 400 all out in 131.1 overs (Rohan Prem 170; Timir Chanda 4/77, Manisankar Murasingh 3/79)
Tripura 1st innings: 55/4 in 31 overs (Unnikrishnan Manukrishnan 4/20).