In front of a reasonable crowd at Edgbaston, Hampshire won the toss and opted to bowl first.
Neil Carter lasted only two balls before he was bowled for one by Dominic Cork and Keith Barker went the same way bowled by Briggs for 28.
Aften ten overs of a match shown live on Sky Sports, Warwicks, without Jon Trott who was on international duty, were 79-2 and then Darren Maddy was the third man out, hitting the ball high into the air and being caught by Vince for 44 from 31 balls.
Jim Troughton and Ian Westwood came together and Troughton was given run out by tv umpire Martin Bodenham for 12 off the bowling off Cork to leave Warwicks 88-4 after 11 overs.
The underfire Westwood, who captains Warwickshire's ailing County Championship side, gave his wicket away with an easy catch to Wood short of the boundary to give young Danny Briggs his third wicket.
Warwicks passed the 100 mark after 13.1 overs with Rikki Clarke and Tim Ambrose at the crease.
Rikki Clarke smacked a six in the 16th over, the first boundary since Maddy was out. The pair put on 58 to see Warwicks finish on 153-5.
Neil Carter bowled the first over for a Warwickshire side which had Calum McLeod fielding for Ian Westwood who had twisted his knee. Five runs came off the first over and 29 more were added before Chris Woakes caught Jimmy Adams.
Hampshire were 37-1 after five overs, two runs ahead of Warwickshire score at that time.
The fifty came up with a six for Razzaq and by the end of the eighth over, they were 65-1. Razzaq was out for 33 bowled by Stefan Piolet who also bowled McKenzie to make the score 101-3 after 12.2overs.
James Vince was making steady progress and although his partners Ervine and Carberry lost their wickets, Vince was there at the end on 66 not out as
Hampshire reached the total with a ball to spare and Warwicks out at the Quarter-Final stage for the sixth successive time.
Warwickshire are next in action on Sunday with a Pro40 game against Scotland at Citylets in Edinburgh. Play is due to start at 1.45pm.