The reigning champions needed only 136 for their sixth win of the season, but they slumped to 113 for 8 before Dougie Brown (26 not out) and Ashley Giles (10no) put on 23 in six nerve-wracking overs.
Brown's winning boundary put Warwickshire within 38 points of Derbyshire with a game in hand - and the teams are still to meet at Derby, starting on 12 September.
Warwickshire also still have to play Surrey and Essex, and Neale said: "Following our victory over Durham last week this was a significant win. It does not mean a huge rise in the table but it keeps us very much in contention. We are closing on the teams above us and we can do ourselves a lot of good by beating three of those sides in our last five games."
Robin Smith (77) and Will Kendall (75) gave Hampshire a hard-earned draw against Lancashire at Old Trafford.
Hampshire never threatened to reach their victory target of 415 after resuming on 21 for 1, but comfortably reached safety at 304 for 6 before Mike Watkinson, the Lancashire captain, called a halt with three overs remaining.
Middlesex held on for a draw against Worcestershire in a thrilling finish at Lord's which saw them finish only nine runs short of their victory target of 251, but with just one wicket in hand.
They were given a superb start by Paul Weekes and Jason Pooley, who put on 100 for the first wicket, but started to waver after losing their captain, Mike Gatting, first ball.
Vikram Solanki, who bowled Gatting, returned career-best figures of 5 for 69 as Middlesex were finished on 242 for 9 off 49 overs.Reuse content