The visitors were 140/5 from 59.4 overs after day two, leaving the home team frustrated and it would now be seeking three points by virtue of a first-innings lead.
Pragyan Ojha (3/30) and Ashok Dinda (1/28), who gave the first breakthrough, took one wicket each on a day when only 65 minutes of action was possible and the visitors added 38 runs to their overnight total.
Earlier in the morning, the match began after a 90-minute delay with Rajasthan 102/3 from 43.2 overs as only 3/4 overs could be bowled before the skies opened up.
Play resumed at 1.30pm in the second session and the Bengal pacer Dinda gave the first breakthrough in the fourth over of the day when Puneet Yadav (14) edged one to Sudeep Chatterjee at the gully.
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The former India spinner (3/30) was once again the pick of the lot but to Bengal's frustration, rain interrupted proceedings again at 2.20pm and that was it for the day.
Brief Scores:
Rajasthan 140/5; 59.4 overs (Vineet Saxena 44 batting, Ashok Menaria 40; Pragyan Ojha 3/30) vs Rajasthan.