Sent into bat, Mumbai scored 156 for six from 17.5 overs, riding on Kieron Pollard's counter-attacking 20-ball unbeaten 37 and opener Dwayne Smith's identical score of 37 before rain came down in the Group B match at the Newlands.
The umpires waited for one and half hours for play to resume before finally calling off the match.
Both the sides got two points each from today's match and Mumbai, who had lost their opening match against Highveld Lions, would now need to win all their remaining two matches for any chance of qualifying for the semifinals.
For Mumbai, it was a stop-start performance after they were sent into bat with wickets falling at regular intervals before Pollard threatened to cut loose at the later stage.
Pollard hit three fours and two sixes from the 20 balls he faced before rain came to stop the Mumbai innings.
None of the Mumbai batsmen could score big individually and the highest partnership was the 36-run stand between opener Dwayne Smith (37) and Rohit Sharma (25) for the second wicket.
The players came back into the field twice only to leave it immediately due to rain before the umpires ended the innings. When the first rain break came after the completion of the 17th over, Mumbai were 145 for six and Pollard hit a six from one of the five balls that were bowled before another rain break to take his side to 156 for six.
For Yorkshire, spinners Adil Rashid and Azeem Rafiq grabbed two wickets apiece while Joe Root took a wicket.
Earlier, Smith gave a dramatic start to the Mumbai innings as he smashed off-spinner Root, who opened bowling for Yorkshire, for two consecutive sixes from the second and third balls of the match.
