Darren Maddy had made 16 of the 44 runs when he was first out. That was in the seventh over and in the next over, Keith Barker was out without scoring.
Jim Troughton added ten before Carter and Westwood made a fourth wicket stand of 112. Westwood was caught by Pope after making 47 but with the score on 191, Neil Carter who has just passed the one huindred mark and hit 17 fours in 93 balls was out for 101.
Ant Botha added 42 as Warwicks ended their forty overs on 272-8.
When Leicestershire batted, Neil Carter bowled du Toit in the 6th over and six overs later, his fellow opener Benning was out for 46 bowled by Boyd Rankin.
Tight bowling restricted Leicestershire to 95-3 at the half way mark and despite James Taylor hitting an unbeaten 103, Warwicks ensured the run-rate stayed below the required run rate to see the home side finish on 247-6 and give The Bears a 25 run victory.
Tomorrow, Warwickshire are at home to Hampshire in the Twenty/20 Cup Quarter-Final. Play is due to start at 16.40 and admission is £20 advance £25 on the gate but is cheaper for members.
You can hear the match live on the BBC Coventry & Warwickshire website.
Jon Trott has been named in the England team to face Pakistan on Thursday.