At stumps in the Group A match, the hosts were 20 for one, with opener Jiwanjot Singh back in the hut at the PCA stadium.
Punjab now trail the visitors by 202 runs with nine wickets remaining in the first innings.
Medium-pacer Kaul returned impressive figures of four for 63 while Sandeep Sharma finished with three for 72. There were two wickets for Baltej Singh.
Puneet Yadav topped the scoring for Rajasthan with a knock of 57 that came off 74 balls, and was studded with 11 hits to the fence.
After being dealt with a few early blows, Rajasthan seemed to have recovered with a half-century stand between Yadav and Ashok Menaria (32), but the former was trapped in front by Sharma in the 30th over.
Rajesh Bishnoi made 23 even as Deepak Chahar took the visitors past 200 with a handy 35.
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Karan Goel and Ravi Inder Singh were batting on 11 and seven respectively when the stumps were drawn.
Breif scores:
Rajasthan 1st innings: 222 all out in 70.3 overs (Puneet Yadav 57; Siddarth Kaul 4/63, Sandeep Sharma 3/72).
Punjab 1st innings: 20 for no one in 20 overs.