The visitors, who won the toss and elected to bat, scored well early on before losing opener Mayank Agarwal for just one in the third over.
In response, the Sunrisers roared to 50 from six overs, before taking a strategic timeout and losing Shikhar Dhawan (18) on the first ball after the restart, and Warner (28) just two balls later.
A spell of economic bowling from the double leg spin attack of Imran Tahir and Amit Mishra, and a disciplined performance with the ball from Matthews, restricted the home side further, with Lokesh Rahul (24) and Naman Ojha (12) posting modest scores.
However, the steady batting of Ravi Bopara (41) kept the Sunrisers in contention until he was caught by Manoj Tiwary, and when Eoin Morgan was clean bowled by Duminy with the following delivery, the fightback appeared to be over.
A late rally by Karn Sharma (19) provided some drama as back-to-back sixes left the Sunrisers needing just 10 off the final over to win, but Ashish Reddy's (15) run out with three balls remaining allowed the Daredevils to wrap up the win, with the Sunrisers finishing on 163-8.