Klinger, who is set to call time on his 21-year career at the end of the tournament, reached his eighth T20 century from the final ball of Gloucestershire's innings as the visitors posted 180 for three.
Only Chris Gayle has scored more centuries than the 39-year-old Australian in the format – with 21 – but his team were still made to sweat for victory as Faf du Plessis threatened to overhaul the target on his Spitfires debut.
The South Africa captain shared a 122-run stand with Sam Billings but both fell to Ryan Higgins, who claimed the key wickets of Alex Blake and Du Plessis from consecutive deliveries to unravel the pursuit.
While victory also almost certainly assured Gloucestershire of a home quarter-final next week it left Kent in a five-way battle to for the remaining two last-eight places, which will be decided on Friday night.
Essex and Hampshire ensured they remained amongst that group after keeping their hopes alive with victories against Surrey and Middlesex respectively.
Veteran Eagles pair Ravi Bopara and Ryan ten Doeschate shared an unbroken 117-run stand to set-up a 19-run win that ended Surrey's hopes of progressing at the Kia Oval.
Jamie Porter claimed four wickets for 38 runs, including Surrey's top three, and although Jordan Clark hit 60 from 32 balls Surrey were bowled out for 167.
Hampshire skipper James Vince hit an unbeaten 66 from 55 balls at the top of the order and shared a 96-run stand with Aneurin Donald (51) to overhaul Middlesex's 153 for five.
Victory moved Hampshire up to fifth, ahead of their final night meeting at winless Glamorgan, as they look to force their way past Kent or Somerset in the South Group's top four.
Kent can book their progress with a win at Essex on Friday night while Somerset and Middlesex – locked on 13 points with Hampshire and Essex – face a virtual elimination match at Taunton.
Yorkshire climbed off the foot of the North Group table with a comfortable 80-run win over Northamptonshire at Headingley.
Left-armer David Willey claimed four wickets against his former county after home openers Tom Kohler-Cadmore and Adam Lyth hit half-centuries to pave the way for Yorkshire's 187 for seven. Willey finished with four for 18 as the Steelbacks were bowled out for 107.