Batting first, the Delhi outfit were struggling after a brace of wickets from the Australian pair Shaun Tait and Shane Warne, before Warner (54 off 44 balls) and Rao (60 off 40) compiled an 83-run stand to reach 151.
In reply, veteran Rahul Dravid (38) provided a solid foundation and despite losing four wickets, Johan Botha's unbeaten 39 off 32 deliveries and Ross Taylor's match-wining six saw Royals to a relatively easy win.
Daredevils lost captain Virender Sehwag (four) in the very first over to Tait, who also had Aaron Finch (11) caught at mid-on, while Royals skipper Warne compounded the misery by dispatching Unmukt Chand and Naman Ojha within his first three balls.
Reeling on 36 for three, Warner and Rao eased Delhi's nerves with their 61-ball salvo, striking 10 boundaries and five sixes between them before falling to Amit Singh and Siddharth Trivedi respectively.
Irfan Pathan (nine not out) and Roelof van der Merwe (two not out) then led the Daredevils into the break - the former hitting two boundaries to round things up.
The Royals' start was in sharp contrast to the one made by Delhi - despite the early wicket of Amit Pauniker - thanks to a quick-fire Dravid innings that included six boundaries.
His dismissal brought in Ashok Menaria, who smashed three sixes for his 15-ball 22 but fell to Pathan. Botha had helped build a useful 29-run stand with Ajinkya Rahane (18) but Morne Morkel gave Delhi hope by snaring the latter.
However, with 31 required off five overs, Botha's largely patient approach was enough to seal victory with nine balls remaining, as Taylor (18 not out) finished things off with a six.