CSK reached the target of 173 with two balls to spare with captain Mahendra Singh Dhoni remaining not out on 18 off six deliveries.
Ruturaj Gaikwad top-scored with a 50-ball 70 while Robin Uthappa contributed a 44-ball 63.
For DC, Tom Curran took three wickets while Anrich Nortje and Avesh Khan got one each.
Earlier, Prithvi Shaw top-scored with a 34-ball 60, while captain Rishabh Pant and Shimron Hetmyer contributed 51 not out and 37 respectively after DC scored 172 for 5 after being invited to bat.
Josh Hazlewood was the pick of CSK bowlers with figures of 2/29 while Ravindra Jadeja, Dwayne Bravo and Moeen Ali took a wicket apiece.
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