Chasing a target of 163, Delhi Capitals, who had lost the way after a blazing start, were brought to 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.
Delhi Capital 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 was 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 Powerplay overs.
One of the most regal shots played by Prithvi 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 leg-side to see the end of Dhawan even as Prithvi completed his 50 off 31 balls.
However, just like 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 they 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, 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.