The former champions ended the day at 76 for no loss with wicketkeeper-batsman Aditya Tare unbeaten on 43 and Kaustubh Pawar on 26.
Rajasthan resumed the day on 269 for two, but lost centurion Vineet Saxena early when Tare held on to a snick off pacer Dhawal Kulkarni.
Robin Bisht (34), Ashok Menaria (26) and R R Parida (28) failed to cash in on starts, but wicketkeeper Dishant Yagnik stayed unbeaten on 76 off 123 balls.
It was a tiring day for Mumbai skipper Rohit Sharma, who used eight bowlers to tackle Rajasthan. The day ended on a positive note with his openers playing out 28 testing overs.
Mumbai Roundup: 9 incidents that took place this week
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Photos: Dipika Pallikal Karthik oozes oomph in bikinis and sarees
Photos: Kareena Kapoor Khan's grand birthday bash
Photos: 'Dangal' girls Sanya Malhotra, Fatima Sana Shaikh's dinner outing