They managed to create quite a number of clear scoring chances but their strikers Anderson, Thoi and Farukh Chaudhary were unable to convert.
Lanky medio Karan Sawhney had a couple of attempts but he also did not succeed in finding the net.
Chennai's goalkeeper Karnajit Singh came under a lot of pressure but made some timely to go into the breather with a clean slate.
The Mumbai club continued to hold the upper hand after resumption and in the very first minute substitute Alex Silva, with a precise flick, created a clear chance for Thoi but Karanjit rushed out and foiled the attempt.
Thoi got back possession but his attempt was thwarted by defender Abhishek Das.
Chennai had just one good chance in the second half when their Brazilian striker Charles DeSouza broke through down the right but his flick bounced off the near post and hit Yumnam Raju on the face.
Mumbai will now host Shillong Lajong FC in their penultimate round match on April 23.