Jammu and Kashmir's performance on the fourth day was built around Parvez Rasool, who finished with 7 for 87 in the second innings to restrict Tripura to 272.
The hosts began the day at 103 for 2, and the overnight pair of Abhijit Dey and Kaushal Acharjee added another 48 runs before Dey fell to Rasool.
The Jammu and Kashmir team was in headlines last week as the Jammu and Kahmir Police raided the hotels of Jammu where the state Ranji team was staying.
In 2000-01 Jammu and Kashmir played a pre- quarterfinal clash against Odisha.
After Goa's victory over Andhra Pradesh on Wednesday, they were tied with Jammu and Kashmir on four wins from eight games.
Having already secured three points from the first innings lead, Jammu and Kashmir needed to beat Tripura on the last day or ensure they kept their score in the second innings above Goa's quotient to reach the next round of the tournament.
Congratulating the team, Jammu and Kashmir Chief Minister Omar Abdullah tweeted: Congratulations to the J&K cricket team for reaching the quarters of the Ranji trophy. 1st time since 2000 so well done chaps."
Jammu and Kashmir will now play Punjab in the quarterfinal at Moti Bagh in Baroda from Jan 8.
--IBNS (Posted on 02-01-2014)