Graham Onions picked up two quick wickets early on in the morning session, with Clarke out lbw on 14 within 20 minutes of play starting and Troughton caught behind, leaving Warwickshire 67-6, with a lead of only 31.
Scott Borthwick claimed his fourth wicket of the innings, making this his career best first class performance, as Best, the nightwatchman, was given out lbw having contributed 2 to the total.
The fifth delivery from Ruel Brathwaite saw Woakes bowled outright for 11 and Warwickshire slumped to 93-8, with a lead of just 57.
At lunch the visitors were 94-8 with Rankin and Ambrose, who went on to get fifty but was dismissed shortly after when trapped lbw by Onions, just as Warwickshire had extended their lead to 100.
Rankin was the final man to fall, when he became Onions fourth wicket of the innings after a fine catch from Callum Thorp.
Durham were set 102 to win with 49 overs remaining.
Will Smith opened alongside Scott Borthwick, who was moved up the batting order to replace Michael Di Venuto, who could bat no higher than seventh due to the fact he hadn’t fielded due to sustaining a thigh injury in the first innings.
Westwood dived almost backwards from deep square leg to hold the catch off Woakes which sent Will Smith (9) back to the pavilion.
Muchall (30) was caught by Troughton off Best, with Durham requiring 19 runs, which brought Ben Stokes to join Scott Borthwick, both of whom saw Durham to the end of play with the latter hitting the final boundary which secured the home side victory.
Durham claimed victory by 8 wickets with Borthwick (48*) and Stokes (9*) the unbeaten batsmen.