Chasing 177 runs to win, Karthik and Ferveez Maharoof held their nerves to knock off the required eight runs in the last over of the match with a ball remaining.
Karthik finished unbeaten 56 off just 36 balls and his innings was studded with four hits to the three sixes while Maharoof gave him good company and remained not out on 20. Both the batsmen added 49 runs for the sixth wicket.
With this win, Daredevils can still hope for a semi-final berth and would keenly follow Mumbai Indians' next two matches. Delhi can reach the semis if Mumbai Indians lose any one of their remaining two matches.
Virender Sehwag made his intentions very clear from the start as he blasted away 19 runs from the very first over off Shaun Pollock which also included a huge hit over the mid-wicket fence.
But the swashbuckling opener could not continue his start as a delivery from Ashish Nehra uprooted his leg stump. He contributed 18 runs from just six balls.
Gambhir then took charge from Sehwag and looked to tear apart Andre Nel and launched 11 runs off South African recruit's opening over.
But Nel had the last laugh when Gambhir lofted the bowler high only to see Dwayne Smith complete the catch after a long wait.
Gambhir raced to a nine-ball 19.
Shikhar Dhawan (27 off 29 balls) and Manoj tiwary then tried to stitch the Delhi innings and mid-way through the innings were at 88 for two.
But Dhawan, just when looked set to provide the home team the required momentum, was grabbed by Siddharth Chitnis at the long-on boundary off Dwayne Smith.
Tilakratne Dilshan could not do much and was clean bowled on zero by Smith off the very next ball to make it 89 for four.
Tiwary then looked to justify his selection and along with Dinesh Karthik combined 41 runs for the fifth wicket. Karthik, in particular, was looking in good touch as he scored 27 runs in the 41-run stand.
But a horrendous mix-up in the middle saw Tiwary's back as both the batsmen were left stranded at the same end. Tiwary understood the need of the hour and made a well-compiled 36 off 29 balls.
Karthik, however, was unfazed by the wickets tumbling at the other end and was going from strength to strength. He completed his half century off 27 balls with a stright six off Nel as his team was smelling victory.
Earlier, set to bat first, Sanath Jayasurya and Sachin Tendulkar gave a rollicking start to Mumbai Indians and utilised the power-play period to good effect as they raced to 60 in their first six overs.
Jayasurya launched some astonishing shots over the ropes while Tendulkar was content to witness the left-hander's blitzkrieg from the other end.
Brett Geeves, who made his IPL debut for Delhi today, did not have an auspicious start as both Tendulkar and Jayasurya took 15 runs from his first over but his replacement Yo Mahesh too was clobbered for 17 runs in his opening over.
Just when Sehwag's decision to field first appeared to backfire, Yo Mahesh produced a peach of a delivery which nipped back and took away Tendulkar's leg-stump.
Tendulkar scored 18 off 16 balls with the help of three well-timed deliveries in the 70-run stand for the opening wicket.
Meanwhile, Jayasurya continued his attack and completed his half century in just 30 balls which was studded with five hits to the fence and four sixes.
Half way through the innings, Mumbai had reached 81 for one with Jayasurya looking threatening for Delhi who were eyeing a semi-final berth in their last league match tonight.
A huge six in to the long-on stands off leg-spinner Amit Mishra from Jayasurya raised Mumbai Indians' 100 in the 12 over.
However, Daredevils managed to make a comeback with some quick wickets and Jayasurya's wicket, when he was on 66, must have been the most satisfying one for the hosts.
Jayasurya became a victim of Mishra's flight and while trying to loft the leggie to long-off fence fetched the waiting hands of Tilakratne Dilshan. By then, Jayasurya had laid a good foundation for the late-order batsmen to go for the onslaught. The left-hander's 66 came off just 42 balls with five effortless sixes along with four boundaries.
For Daredevils, Yo Mahesh was the most successful bowler and took four wickets, two of which came in his last over, to finish with figures of 36 runs from four overs.
Barring Jayasurya, the other batsmen could not make any major contributions and fell while trying to increase the tempo.
While Geeves emerged the most expensive bowler for Sehwag giving away 50 runs in his four overs, Mishra took the important wicket of Jayasurya to finish his four overs for 23 runs.
Dwayne Smith made run-a-ball 17 while Abhishek Nayar (1) became the only victim of Geeves.
South African all-rounder Shaun Pollock, too, failed to make a mark with the bat and was bowled off Ferveez Maharoof on when he was on 16.
Robiin Uthappa, who has not had a great tournament so far, decided to take on the pace spearhead Glen McGrath and launched three successive boundaries in his last over.
McGrath produced his worst bowling performance of the tournament as he went wicketless in four overs giving away 39 runs.
Uthappa remained unbeaten on 46 off just 33 balls which included five fours and two sixes and helped his team to post 176 for eight which always seemed a challenging total for the hosts.