The home side batted first at the Rose Bowl and Chris Woakes struck early taking the wickets of Michael Carberry and Adams.
Boyd Rankin also took two wickets removing Vince who had made 30 and Ervine who had made 35 to see Hants on 78-4.
Imran Tahir had Nic Pothas trapped lbw before Darren Maddy had Dawson caught and Dominic Cork bowled.
Imran Tahir caught Wood off the bowling of Maddy but Neil McKenzie was there at the end of the innings unbeaten on 62 to see Hants close on 211-8.
It was a reasonably low total for Warwicks to get and after Neil Carter was out for 15, Jon Trott and Ian Bell put on a second wicket stand of 155 before Bell, who had faced 72 balls and hit 71 runs including seven fours was caught by Adams.
Jim Troughton came in and made 14 whilst Trott went on to make an unbeaten 96 as Warwicks reached 212 with 4.2 overs to spare.
Warwicks take on Kent in a Pro40 League game on Sunday at Edgbaston. PLay is due to start at 1.45pm.
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