They resumed on 246-9 at Edgbaston and Ant Botha and Boyd Rankine took the score on to 276 before the Irishman was out for seven.
Warwicks were forced to follow on and their supporters witnessed another collapse as Graham Onions took four wickets.
Tony Frost was his first victim out for three with the score on three. This was followed by Ian Westwood falling to Blackwell for eight and Ian Bell who had made 57 out after the one hundred mark was reached.
Jon Trott who had partnered Bell was out a run later for 32. Then Jim Troughton and Tim Ambrose were out cheaply as Warwicks found themselves in big trouble.
Neil Carter tried a rescue act in his own inimitable style by hitting five fours but he was caught by Blenkinstein off the bowling of Steve Harmison for 23.
Onions added two more wickets to his haul as Warwicks were now ahead of Durham's scure but with only one wicket left.
Ant Botha and Boyd Rankine were left with the task of batting for an hour to scrape a draw. It didn't last and Rankine was out for two from 18 balls give Onions figures of 7-38.
With Warwicks all out for 177, this left Durham needing twenty one to win.
Will Smith and Michael Di Venuto batted for Durham and hit the runs inside four overs to win by ten wickets.
Warwicks next match is in the Twenty/20 game against Gloucestershire at Bristol. Play is due to start at 5.30pm and commentary can be heard on BBC Coventry & Warwickshire web-site.