Pandey, who remained unbeaten on 69 off 49 balls rescued the visitors with a brilliant 110-run fourth wicket stand with Pathan (59) to turn the tide in favour of the visitors after a dreadful start.
Needing nine from the last over, Mishra got Chris Woakes stumped off his second delivery but couldn't prevent Pandey from hitting him for a six off the fourth ball and then leaked two runs off the next to lose the tie.
Earlier, skipper Zaheer Khan gave Delhi a perfect start with the wickets of his opposite number Gautam Gambhir (14) and Colin de Grandhomme (1) while Pat Cummins sent back one down Robin Uthappa (4) to reduce the visitors to 21/3.
Zaheer struck in the opening over of the innings to remove KKR's debutant de Grandhomme, who holed out a short of length delivery to give a simple catch to Sam Billings at deep square leg.
The next over saw Cummins getting belted for two boundaries by Gambhir and one from Uthappa before the Australian hit back with the wicket of the KKR stumper on the last ball of the over.
The third over witnessed the fall of an in-form Gambhir, who found a leading edge from Zaheer to be smartly latched on by Angelo Mathews running back a few steps at backward point. The southpaw faced 12 deliveries and fired two boundaries.
Thereafter, Yusuf Pathan (59) and Manish Pandey batted sensibly to resurrect the visitors' innings with a 110-run fourth wicket stand, as KKR went past the 100-run mark with ease in the 13th over.
Pathan, expectedly was the more aggressive of the two with the Baroda right-hander slamming two consecutive boundaries off Cummins before lofting Mohammed Shami for a huge one-handed six over long-on.
The 34-year-old Pathan brought up his first half century of this year's IPL in style by dispatching Zaheer for a flat six over long on while completing the 100-run stand off 66 deliveries.
Pandey, at the other end, played the perfect second fiddle, by hitting the occasional boundaries to get to his half century off 37 balls.
Just when the match seemed to be slipping out of Delhi's hands, South African all-rounder Chris Morris provided some respite to the capacity crowd at the Kotla with the wicket of the dangerous Pathan.
Pathan, who had by now faced 39 deliveries and struck six boundaries and two maximums, got a leading edge off a quick delivery from Morris, who took it on his own follow through.
The fall of Pathan, brought in Suryakumar Yadav (7), who chipped in with a 21-run fifth wicket stand with Pandey before Cummins dismissed him on the penultimate ball of his fourth over.
In the end, Pandey kept his cool, striking four boundaries and three sixes to help the visitors get over the line.
Earlier, promising wicketkeeper-batsman Rishabh Pant's late surge helped Delhi Daredevils reach 168/7 as the hosts squandered a promising start by the openers after electing to bat.
Delhi rushed to a 50-run opening stand in just 4.4 overs with the opening duo of Sanju Samson (39) and Sam Billings (21) sending the KKR bowlers on a leather hunt.
Samson's purple patch continued to help the home side, with the Kerala youngster slamming seven boundaries, including three off pacer Umesh Yadav's over to lay the platform for a massive total.
Billings looked good during his brief 17-ball stay, helping the ball to the ropes on two occasions before being caught behind by Robin Uthappa off pacer Nathan Coulter-Nile.
With Delhi at 63/2 after the eighth over, Shreyas Iyer (26) and Karun Nair (21) joined forces to add 43 runs for the third wicket before the Mumbaikar was undone, in his attempt to steal a second run, by a superb throw from de Grandhomme straight on the stumps.
KKR made a brilliant comeback after the initial onslaught, to stop the flow of runs in the middle overs before Pant (38) took the attack to the opposition in the 17th over from Umesh.
Pant slammed four mammoth sixes, including a brilliant flick down the legs towards square leg, besides two boundaries as Delhi neared the 150-run mark.
Morris then struck two consecutive fours off Woakes to take Delhi to 168 with a 9-ball 16.
Being five down inside first 6-7 overs, was always going to be difficult, says Gautam Gambhir
We deserve to be in the final ,says MI skipper Rohit Sharma
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