Chasing a target of 163, Delhi Capitals, who had lost the way after a blazing start, were brought to the cusp of victory by young Pant, who smashed 49 off 21 balls with five sixes as West Indies' Paul finished it off the penultimate ball of the innings with a boundary off Khaleel Ahmed.
Pant, who resurrected the innings from a precarious 111 for 5 couldn't have chosen a better venue than the chairman of selectors MSK Prasad's home ground to drive home a point on his much-debated World Cup exclusion.
It was a sense of deja vu for Paul, who had similarly hit winning runs in the U-19 World Cup final in 2016 when Khaleel was the bowler in the final over.
Delhi Capitals now face Chennai Super Kings in the second Qualifier in their bid to make it to the maiden IPL final against Mumbai Indians.
Needing 34 from three overs, Pant showed brilliant game awareness to target Sunrisers' weakest link Basil Thampi, smashing him for 21 runs in the 18th over as he brought the game on even keel.
A six off Bhuvneshwar Kumar brought the equation down to five runs before Pant in his bid to finish the game holed out with five required from seven balls.
When the chase started, Prithvi Shaw (56 off 38 balls) was in his element in the fifth over bowled by Bhuvneshwar Kumar when he first played a flowing cover drive, followed by a ramp shot for six over third man and a lofted boundary over mid-off to put Sunrisers on the backfoot.
Sensing that Prithvi has taken the lead, senior partner Shikhar Dhawan (17) after collecting three boundaries started playing the second fiddle with 55 coming off the Powerplay overs.
One of the most regal shots played by Prithvi Shaw was a six over long-on when he gave off-spinner Deepak Hooda the charge.
In the same over, Wriddhiman Saha effected a smart stumping down the leg-side to see the end of Dhawan even as Prithvi completed his 50 off 31 balls.
However, just like the Sunrisers, Delhi also lost momentum at the start of the back-10 with Khaleel's extra pace and bounce accounting for Prithvi and skipper Shreyas Iyer (8). Prithvi's innings had six fours and two sixes.
It was Rashid Khan (2/15 in 4 overs), who then removed Colin Munro (14) and Axar Patel (0) to peg back Delhi at 111 for 5.
Pant, who teed off with a six off Mohammed Nabi, hit him for a second six as 42 runs were required off the last four overs when Pant changed the match.
Earlier, the Delhi Capitals bowlers pulled the plug in the middle overs, restricting Sunrisers Hyderabad to a par score of 162 for 8 on a good batting surface after opting to field.
Keemo Paul (3/32 in 4 overs) was ably supported by Amit Mishra (1/16 in 4 overs), Ishant Sharma (2/34 in 4 overs) and Axar Patel (0/30 in 4 overs) as the quartet bowled 40 dot balls between them.
Ishant got rid of Wriddhiman Saha (8), who offered a simple catch to Shreyas Iyer at mid-off.
Martin Guptill (36, 19 balls) hit three sixes including two off his countrymate Trent Boult to set the pace in the Powerplay overs which yielded 54 runs.
Credit to the Delhi bowlers as the next 14 overs produced only 108 runs.
As Guptill was in an attacking mood, it gave the normally flamboyant Manish Pandey (30, 36 balls) an opportunity to play around before the spinners came in operation.
Axar and Mishra choked the run-flow and Guptill was accounted for trying to slog sweep the leg-spinner. His innings had four sixes apart from a boundary.
Sunrisers suddenly lost momentum in those middle overs as Pandey and Kane Williamson (27 off 26 balls) added only 34 runs in seven overs before the Karnataka batsman was dismissed by a Keemo Paul slower delivery. Pandey was dismissed by Sherfane Rutherford at the long-on boundary making it 90 for 3 for the Sunrisers.
Ishant coming in his second spell bowled the perfect inswinging yorker to send back Williamson.
Vijay Shankar (25 off 11 balls) and Mohammed Nabi (20 off 13 balls) added 36 runs in 2.4 overs to beef up the total but Paul kept things tight in the end.