Chasing a victory target of 166, Delhi batsmen failed to get 17 runs off the final two overs to finish at 160 for six at the Feroze Shah Kotla.
David Warner's smashing innings of 77 went in vain as the home team lost their second match in a row, having beaten by Kolkata Knight Riders in their tournament opener.
Needing nine runs from the final over, Cooper bowled a tight line conceding only three runs as Delhi lost the wicket of Andre Russell and Johan Botha.
A six off the final delivery was all that Delhi required but Cooper kept it fuller as Naman Ojha failed to connect the ball.
The match, which seemed to have slipped out of the hands of the Royals, was turned on its head by Australian Brad Hodge whose brilliant piece of fielding led to the dismissal of Warner.
Warner, who till then looked like anchoring the Daredevils chase to perfection, was run-out in the penultimate over as he failed to beat a direct throw from Hodge who was manning the cover region. Such was the effect of that dismissal that Daredevils completely lost the plot during the final nine deliveries of the match.
Warner hit nine fours and a big six off Siddharth Trivedi in his 56-ball knock. He shared an important 67-run third wicket partnership with young Manprit Juneja (20) as the home team did not look like facing any major hiccups in the chase.
But, when it mattered the most, unheralded Rahul Shukla and Cooper did simple things admirably well to help their team start off on a winning note.
Warner got off to a sluggish start, taking 12 deliveries for his first four runs but as the innings progressed he opened up to play some smashing shots. He got a couple of reprieves on individual scores of three and 57 as Rahul Dravid and Rahul Shukla dropped him.
Delhi started their run chase cautiously as Warner shared an opening stand of 39 with promising opener Unmukt Chand (23 off 34 deliveries). Mahela Jayawardene (19) got out cheaply as Ajinkya Rahane pulled off a stunner at point to dismiss the Delhi batting mainstay off the bowling of pacer Rahul Shukla.
For Rajasthan, Cooper returned with figures of three for 30 in his four overs while S Sresanth (1/18) and Rahul Shukla (1/27) got one wicket each.
Earlier, the Royals rode on skipper Rahul Dravid's half century to post 165 for seven after opting to bat on a flat Kotla track.
Dravid (65) was ably supported by all-rounder Stuart Binny (40) with whom he stitched a 55-run partnership in just 4.4 overs to take the Royals to 165 for seven.
Dravid's knock came off just 51 deliveries smashing six fours and two sixes while Binny took just 20 balls for his thrilling cameo as he played some breathtaking shots during his stay at the crease. He hit three sixes and two boundaries before being castled by Umesh Yadav.
Rajasthan were looking set for a bigger total but their march was halted by Delhi quicks Yadav and Ashish Nehra, who bowled well at the death giving away just six runs in the final two overs of the innings.
Yadav took four wickets in as many overs giving away just 24 runs, while Nehra picked two scalps to dent the Royals surge. They visitors lost four wickets in the space of eight deliveries with the addition of only four runs.
Making his debut in the IPL, Sri Lankan wicketkeeper Kusal Perera (14) came out to bat with an attacking intent and smashed the second ball of the innings for a boundary. He was though handed a life by Johan Botha when he was dropped for five in the slips but failed to make the hosts pay.
At the other end, Rahane's first run came off an exquisite on drive to the mid-wicket fence as he got off to a cautious start. In his brief 28-run knock, Rahane gave a glimpse of his last year's form when he hit Andre Russell first over his head followed by a deft touch to the third-man boundary, in two successive deliveries.
Rahane was dismissed by left-arm spinner Shabaaz Nadeem, before adding 65 runs for the second wicket along with Dravid.
Australian Brad Hodge (5) and West Indian all-rounder Kevon Cooper (2) failed to make an impact as they perished in pursuit of quick runs.