The visitors batted first at Edgbaston and had made 43 in seven overs when Boyd Rankin bowled Blackwell for 22.
Although Phil Mustard was the main stay of Durham's innings making 61, wickets tumbled on a regular basis with three for Darren Maddy and two each for Boyd Rankin, Keith Barker and Imran Tahir.
Warwicks restricted Durham to 200-9 in forty overs and Jon Trott got to work on the Durham bowlers after Neil Carter had been removed for a golden duck and Ian Bell was out for thirteen.
Trott hit 13 fours from 98 balls in a superb knock and was on 103 before he was run out.
Jim Troughton had been with Trott and he carried on to make an unbeaten 66 from 76 balls along with Maddy who was not out 13 to see Warwicks home by seven wickets and with five overs to spare.