One-down batsman Rayudu, whose previous highest score this season was 48, made 59 with four sixes and two fours in 43 balls before he was out in the 19th over.
Rayudu's stand of 61 in 54 balls with opener Lendl Simmons (38), playing his first game after being inducted into the squad in place of injured Jalaj Saxena and replacing Ben Dunk in the eleven, laid the platform after the early loss of opener Chidambaram Gautam for 9 in the third over.
Mumbai made a decent start despite losing opener Gautam cheaply. The home team scored on 42 for 1 in the powerplay period of six overs. Simmons carted fellow West Indian Samuel Badree for two sixes over in the sixth over.
But, Mumbai scoring rate slumped later as they could add only 12 runs in the next three overs. The hosts accounted for 33 dot balls in the first nine overs at the end of which the score read 54 for 1. It became 57 for 1 after 10 overs.
Simmons departed when he tried to give charge to R Ashwin and was caught at long on for a run-a-ball 38 that included two sixes and three fours. The stand between him and Rayudu was worth 61 off 54 balls.
The next five overs produced 42 runs leaving Mumbai at 99 for 2 after 15 overs. Though Rayudu carted Jadeja for another six over long on, things had been kept tight by CSK bowlers.
Rohit and Pollard were both caught in deep mid wicket by Suresh Raina to help Ashwin finish with 3/30. Rayudu fell the next over off Mohit and Mumbai were struggling at 129 for 5.
A couple of boundary hits by Tare against Mohit Sharma in the penultimate over and two sixes struck by Anderson off Ishwar Pandey in final over helped Mumbai finish with a 150-plus score.
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