Bhatia's blistering innings off 71 balls helped Rajasthan chase down the 302-run target in 49 overs.
Rahane, playing his second match of the domestic competition, steered Mumbai to 301 for nine.
Besides Bhatia's brilliant effort, Ashok Menaria came up with a 80-ball 86 to contribute to the team's cause.
The other contributions came from Puneet Yadav (49) and Dishant Yagnik (40).
With the loss, Mumbai dropped to the third spot in Group A with 12 points while Rajasthan are fourth with 12 points due to an inferior net run rate.
Mumbai: 301/9 in 50 overs (Ajinkya Rahane 114; Pankaj Singh 4/55).
Rajasthan: 305/5 in 49 overs (Rajat Bhatia 94 not out, Ashok Menaria 86; Abhishek Nayar 2/28).