A rapid half-century by Darren Maddy and three wickets in four deliveries by Chris Woakes overpowered the Unicorns, as Warwickshire won by 60 runs at Edgbaston.
The county side kept up the pressure on Group C leaders Sussex by making 228-9 and this proved to be beyond the part-timers despite an impressive knock by veteran captain Keith Parsons.
The former Somerset player, now 39 and playing Minor Counties cricket for Cornwall, made 58 from 77 balls and put on 71 with ex-Leicestershire wicketkeeper Tom New (31) as the Unicorns replied with 168-7.
Warwickshire's patchy batting display was initially underpinned by Varun Chopra's measured progress to 69 and then set alight by Maddy's first half century in the competition for two years.
The 38-year-old all-rounder settled in during a stand of 57 with Chopra and despite losing a string of partners during a spirited response by the Unicorn's bowlers, he accelerated to an unbeaten 79 from 56 balls.
Warren Lee, a pace bowler who played two List A games for Kent in 2009, gave the visitors an early lift with wickets in successive overs. William Porterfield (24) put up a simple chance to mid-off and Jim Troughton drove the first ball he received to extra cover.
The Unicorns' rotation of their left-arm spinners paid off to an extent with Tim Ambrose (28) lbw to Bradley Wadlan's second delivery and Chopra bowled in the first over of a second spell by Luke Beaven.
Former Lancashire seamer Steven Cheetham later removed Rikki Clarke and Woakes in the space of seven balls but the lower order stayed long enough for Maddy to inflict maximum damage. In all he hit eight fours and two sixes
Lee had Keith Barker caught at short third man for a final return of three for 50, and Wadlan (two for 37) bowled a good final over in which Paul Best was stumped and Jeetan Patel run out.
The Unicorns' response was all but extinguished as Woakes blew a hole in the top order in his opening spell of three for 15 in six overs.
It was the ex-county professionals who averted the danger of a complete collapse, although Parsons, who hit five fours and a six, was dropped twice before Warwickshire regained their momentum.
Clarke eventually ran out New from mid-off and Troughton comfortably caught Parsons at deep mid-wicket in the final over from left-arm spinner Paul Best.