Chennai were reduced to 24 for four, effectively 24 for five, because of an injury to Ambati Rayudu.
But opener Ruturaj Gaikwad batted through the innings for 88 not out from 58 balls to help them post 156 for six.
Mumbai then never recovered from 37 for three, closing on 136 for eight, to give Chennai victory by 20 runs.
CSK captained by India legend M. S. With England all-rounder Moeen Ali in their side, join Delhi Capitals at the top of the table, having won six of their eight games. The IPL has restarted in the United Arab Emirates (UAE) having been postponed in May amid a surge of Coronavirus cases in India.
Chennai Super Kings: 156/6 in 20 overs (R. Gaikwad 88 n.o)
Mumbai Indians: 136/8 in 20 overs (M. Tiwary 50 n.o, D. Bravo 3/25). (BBC)