Delport made an unbeaten 109 as the Foxes inflicted a narrow 2-run victory over Notts Outlaws at Trent Bridge.
The opener scored his runs from just 59 balls, hitting nine fours and six sixes as the Foxes scored 203 for five, after being asked to bat first.
Notts, who had already secured top spot in the North Group and a home match in the knock-out phase, lost both Alex Hales and Riki Wessels inside the first five overs of their chase but were powered back into contention by a scintillating blitz from Dan Christian, who made 73 from 32 balls, with five fours and five sixes.
His departure left the Outlaws needing eight from the final over but Matt Pillans held his nerve and finished with the plaudits and figures of three for 24 as the home side closed on 203 for eight, taking the Foxes into the quarter finals for the first time since 2011.
With runs on the board Leicestershire struck two significant blows as Hales fell to Aadil Ali for 17, having hit the spinner for four consecutive fours previously. Wessels hit Clint McKay to cover and when Tom Moores fell to Pillans, the hosts slipped to 44 for three.
Samit Patel, playing his 150th T20 match for the Outlaws was reprieved on 13 when given run out; keeper Luke Ronchi admitted to not having removed the bails cleanly as he gathered a return from the deep.
Delport picked up the wicket of Brendan Taylor and when Patel fell for 39, half the side had fallen for 100.
Christian swung the bat, to good effect, and put his side in with a real chance when he took 20 off the 17th over but it was Delport who ended his fun, taking a juggling catch on the midwicket boundary.
Earlier, Notts restricted the visitors to just 48 for one in the opening powerplay, with Ronchi falling cheaply to a fine boundary catch by Hales off Harry Gurney.
Ish Sodhi ended a second wicket stand of 60 by bamboozling Mark Cosgrove into miscuing to short third man for 37. The New Zealand leg spinner then persuaded Colin Ackermann to chip a simple return catch straight back to him.
Delport reached his 50 in the thirteenth over, getting to the landmark from 36 balls and he celebrated by hitting Sodhi for two sixes and a four in the next over.
His hundred, the second of his T20 career came from 59 deliveries and the left-hander went past the 103 he scored for Dolphins against Knights in 2013.
Ned Eckersley helped the 28-year old South African add 99 for the fourth wicket and the pair remained together until the penultimate over when Jake Ball speared one through Eckersley’s defences and into the off stump.
Tom Wells trod on his own stumps as he looked to work Gurney to leg, leaving Delport to walk off to the applause from the appreciative Foxes’ following.
Almost 11,000 saw the contest and the Trent Bridge administrators can look forward to another huge gate when the Outlaws host their seventh quarter final in eight seasons, next week.
Leicestershire Foxes, three times T20 winners, must travel to face the winners of the South Group.