From 76/1, RR went to 103/6 when K Gowtham couldn't deal with one from Shivam Mavi that got big.
Royals skipper Ajinkya Rahane played all around a full ball from Kuldeep to get bowled while attempting a reverse sweep.
Tripathi blasted 19 off the second over of the innings, smacking Krishna for a six and three fours.
In the previous match, Kolkata Knight Riders has recovered from consecutive losses and the squad have posted the season's highest total and defeated Kings XI Punjab record. Both Tripathi (27 off 15) and Buttler (39 off 22) improvised to spoil the KKR bowlers' plan of keeping it tight. Kolkata Knight Riders faced Kings XI Punjab in an early start on Saturday.
A threatening batting lineup headlined by the stunning resurgence of Chris Gayle and the fierce striking of KL Rahul, allied with the craftiness of Andrew Tye and surprise offerings of teenage spinner Mujeed Ur Rahman, had a few weeks ago marked KXIP out as genuine play-off contenders.
For the visitors, a win is the need of the hour and just like the home side, they need to win their next two ties so as to decide their own fate.
KKR are ahead of RR on net run-rate but Kallis said: "Winning will be the priority". Thereby, it not only got two vital points but also improved its net run rate to -0.09. Victory in next two games will take them to 16 points and assure them a spot in the play-off stage, but in case of a loss, they will have to win the final game by a huge margin and depend on results of other matches.
Kolkata Knight Riders moved to third on the IPL table after a spectacular collapse from their opponents, Rajasthan Royals, left them a relatively straightforward 143 to chase at the Eden Gardens. Bottom line is, Knight Riders is well-placed to progress after putting back to back wins. From the 10th over when Buttler went, RR lost a wicket in every over till the 13th.
KKR take on already-qualified Sunrisers Hyderabad away on 19 May while Rajasthan host Virat Kohli's Royal Challengers Bangalore on the same day.
Rajasthan, hunting a fourth win on the reel, looked set for an imposing target on Tuesday evening as Buttler and Tripathi teed off - Tripathi, spilled at slip from the first ball of the match by Nitish Rana, striking four boundaries in succession off Prasidh Krishna before Buttler punished Mavi. At this point, of all the contenders, Rajasthan finds itself in the most hard situation.