The 41-time national champions bundled out the Rest for 306, that was mainly built around Karnataka batsman Karun Nair's 94, in reply to their own mammoth first innings total of 603 to take the lead.
However, to give rest to their bowlers and fielders, Mumbai opted to bat instead of enforcing the follow-on and made 2 for 1 in their second innings at stumps.
A stand of 55 between night watchman Jayant Yadav (48) and Sudip Chatterjee (23) moved the score past the 100 mark before Mumbai struck hard by sending back both as well as captain Naman Ojha to leave Rest struggling at 128 for 5.
It was left to Nair, who batted for 275 minutes and struck 11 fours in his 192-ball knock, to revive the tottering innings with help from Sheldon Jackson (37) and tail-ender Jaydev Unadkat (48).
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The bulk of the wickets were taken by medium pacer Abhishek Nayar (3 for 35) and the two spinners - left arm Iqbal Abdullah (2 for 62) and off spinner Jay Bista (2 for 52).
Mumbai lost Akhil Herwadkar cheaply to off spin bowler Jayant Yadav off the fourth ball of their innings before stumps were drawn with first innings centurion Bista on 1.