Neil Carter took three wickets as the visitors batted first.
Thompson top scored with 39 with Boyce 32, New 27 and Taylor 24 as Leicestershire were restricted to 158-8 in their forty overs.
Warwicks treated the game like a twenty/20 match and John Trott and Neil Carter put on 114 for the first wicket.
Carter was the man out, hitting three sixes and nine fours in an innings of 63.
Jon Trott continued his superb run of form and finished 73 not out from 71 balls which included 12 fours as Warwicks made 161-1 with 102 balls to spare to win by nine wickets.
On Tuesday, Warwicks return to County Championship action with a four day game at Nottinghamshire.
In the Test match at Headingly, Australia won by an innings and 80 runs. England, who had Ian Bell out for 3, were skittled out for 263. The series is now tied at 1-1 going into the fifth and final test.