Earlier, Jos Buttler scored 52 off 36 balls as Rajasthan Royals posted a competitive 175 for 8 after batting first.
Buttler hit three sixes and a four in his knock while adding 77 for the second wicket with Devdutt Padikkal (38 off 26 balls).
Ashwin (30 off 22 balls) and Shimron Hetmyer (30 off 18 balls) also made valuable contributions.
Ravindra Jadeja (2/21) had the best figures for CSK while rookie Akash Singh and young Tushar Deshpande also got two wickets apiece.
Brief Scores: Rajasthan Royals 175 for 8 in 20 overs (Jos Buttler 52, Devdutt Padikkal 38; Ravindra Jadeja 2/21)
Chennai Super Kings: 172 for 6 in 20 overs (Devon Conway 50, MS Dhoni 32; Ravichandran Ashwin 2/25).