Craig Meschede claimed three for 17 as Leicestershire struggled after electing to bat first and were dismissed for just 123 in 19.2 overs.
The hosts then lost Aneurin Donald in the first over but that was as good as it got as Jacques Rudolph (46 not out) and Ingram eased Glamorgan to victory with more than six overs to spare.
The Welsh county are 100/30 third favourites to win the competition with Sky Bet behind Hampshire (5/2) and Nottinghamshire (3/1), who host 12/1 outsiders Somerset in their quarter-final clash at Trent Bridge on Thursday night.
Surrey (5/1) host the Birmingham Bears (8/1) on Friday night with the last place at finals day at Edgbaston on September 2 up for grabs.
After a minute's silence in memory of former Glamorgan spinner Don Shepherd, who died last week at the age of 90, Leicestershire started brightly with openers Cameron Delport and Luke Ronchi taking them to 21 without loss after two overs.
Marchant de Lange bowled Delport for 10 in the third over but that did not stem the tide as Mark Cosgrove joined Ronchi in the middle and they raced to 48 for one before Graham Wagg bowled Ronchi for 29 in the fifth over.
Leicestershire then lost their way as Colin Ackermann, Cosgrove and Aadil Ali perished to leave them struggling on 87 for five in the 12th over.
The wickets continued to tumble as Ned Eckersley sliced to third man off Meschede, Tom Wells gave Ingram a simple return catch and Matt Pillans played on to De Lange.
Clint McKay drove straight to cover before the final wicket pair of Gavin Griffiths and Callum Parkinson contributed 21 before Griffiths was bowled by Wagg to end the innings on 123.
Glamorgan got off to a poor start as Donald drove McKay to Ackermann at mid-off in the first over.
That brought Ingram to the wicket and he was immediately into his stride as he swept Ali to the boundary before launching him over the field for another boundary.
Ingram pulled Griffiths for six and then sent a Pillans delivery to the fence for another boundary as Glamorgan reached 37 for one after the powerplay.
The South African was then dropped by McKay at short third man off Pillans before dispatching the bowler into the crowd.
Rudolph then got into the action, helping himself to three fours in an over from Griffiths.
Ingram reached his half century with a single off Delport before ending the match in style with two sixes and two fours off Wells in the 13th over.
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