Saxena and Suwalka were batting on 54 and 70 respectively as Rajasthan ended the day at 155 for one at the Sawai Mansingh Stadium.
Rajasthan now trail by 81 runs with nine first innings wickets in hand.
Opener Siddharth Saraf (23) was the lone Rajasthan batsman out today, off pacer Munaf Patel.
Earlier in the day, Aditya Waghmode (114) hit a century to take Baroda to 236 all out after resuming the day at 163 for six.
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Rajasthan pacer Deepak Chahar grabbed all the four Baroda wickets that fell today, to finish with figures of 5/58. He had taken one Baroda wicket yesterday.
Brief Scores:
Baroda: 236 all out in 69.1 overs (Aditya Waghmode 114, Deepak Chahar 5/58, Pankaj Singh 4/60).
Rajasthan: 155 for one in 54 overs (Vineet Saxena batting 54, Suryaprakash Suwalka batting 70).