Joey Evison (27 from 24) and Grant Stewart (25 from 15) did their best to raise the tempo only to be pegged back by a handy two-over stint of 2-12 from Tom Aspinwall as wickets continued to fall at the death.
Kent’s tail did enough to clamber above the 150 mark, but Wood returned to wrap up the innings by dismissing Klaassen and last man Nathan Gilchrist in the final over for a total that looked below par.
However, the target began to seem more distant for Lancashire when they swiftly sank to 5-2 in reply, with Phil Salt caught in the deep before a ball from Klaassen hit Luke Wells on the pads and trickled down his back leg onto the stumps.
Livingstone gave his side impetus, slamming Nathan Gilchrist for a leg-side six and then, after being cleared to continue when a short ball veered onto his helmet, struck the next delivery for another maximum.
Although skipper Keaton Jennings also departed, caught at backward square off Klaassen, Ashton Turner proved a capable foil for Livingstone with 22 from 14 until he heaved Evison to deep midwicket.
By then, Livingstone had raced to his half-century from 25 balls, launching former team-mate Matt Parkinson for three enormous sixes in a single over to keep Lancashire firmly on track.
The Spitfires leg-spinner struck twice in the space of three deliveries, dismissing Michael Jones (28 from 17) and Hartley for a duck to increase the pressure on Livingstone.
But the 32-year-old responded with another bout of big hitting to take Lancashire across the line, finishing with a booming drive to the cover boundary off Gilchrist.
