Opener Livingstone shared half-century partnerships for the first, third and fourth wickets with Finn Allen, Jos Buttler and Dane Vilas on an Emirates Old Trafford pitch which gripped.
The Falcons posted a competitive 168 for seven, including Luis Reece’s 59 off 32 and three for 28 for Matthew Parkinson, before the Lightning top order entertained a sell-out crowd of approximately 4,000 to win with three balls to spare.
New Zealand debutant Allen crashed 29 off 13 balls in the afternoon sunshine before Buttler’s 30 off 28 in his first county appearance since Finals Day 2019.
Opener Reece was superb against his former side as Derbyshire recovered impressively from captain Billy Godleman’s four-ball duck in the first over, bowled by Livingstone.
Reece was powerful and inventive, though also lost Wayne Madsen retired hurt with a suspected hamstring injury on the way to 50 for one after five overs.
Madsen was struck by a throw from mid-on as he dived to make his ground.
Left-handed Reece reached 50 off 29 balls with a third six, only to fall lbw reverse sweeping at Danny Lamb - 88 for two in the 11th.
He had earlier hit 20 of 21 off Richard Gleeson’s only over on his return from 10 months out with a back stress fracture.
Leus du Plooy and Alex Hughes contributed 34 and 28, sharing 52 for the fourth wicket as they advanced from 91 for three in the 12th.
But both holed out in the deep off successive Parkinson deliveries in the 18th (143 for five).
Livingstone and Allen shared eight fours and two sixes to lay the platform for the Lightning.
Fynn Hudson-Prentice’s seamers then had Allen caught behind down leg and Alex Davies caught at deep square-leg - 53 for two in the fifth over.
But Buttler hit two of his first three balls for leg-side fours.
He later fell caught at short third-man reverse sweeping at Reece’s medium pacers, leaving Lancashire 109 for three after 13.
But a game which produced some rusty fielding from both sides was sealed as Livingstone, following his 36-ball fifty, accelerated to leave only four needed off the last over.
He hit 10 fours and two sixes and batted through a T20 innings for the first time for Lancashire.