The win, which came on the morning of the fourth day, was dramatic as Durham had just one wicket left at a crucial moment.
On the last day of the match, the hosts started ahead by 213 runs for eight wickets and needed 33 runs to reach the target of 246 runs while Ben Rhine and Matthew Potts were at the crease.
Durham needed 33 runs to chase the target of 246 runs to win, while the visitors needed two wickets to end the 17-match target without a win in the County Championship.
The hosts scored 31 out of the required 33 and Ben Rhine completed a brilliant half-century under pressure. But time came for Durham when George Hill dismissed Matthew Potts with a new ball of his spell second ball.
Two runs were still needed to win. But Brydon Carse came to the crease despite his side injury and scored two runs between the third slip and the gully to give Durham much needed victory.
Durham received 19 points and moved ahead of second-placed Glamorgan in the table, while Yorkshire could not win this season.
On a bright morning in front of the crowd at Chester-le-Street, Yorkshire turned to Hill for the first over of the day, hoping they could turn the match around by dismissing set player Rhine.
Rhine and Potts knew that if they batted well in the first few overs, they could win this match. Keeping this in mind, Durham’s batsmen put on a half-century partnership in 127 balls to lead the hosts towards the goal of victory.
A few overs later, Jordan Thompson came in to bowl in place of Hill, but Potts took advantage of a loose ball to hit a four and reduced the margin of victory runs slightly.
Durham was inching closer to the target but Shan Masood turned the dice and handed the ball to off-spinner Dom Bess. Bess bowled hard but he couldn’t stop Rhine from scoring his 15th first-class half-century.
As soon as the second new ball came on the field, the Yorkshire captain turned to Hill and got instant success. He dismissed Potts for 25 runs while only two runs were needed to win.
Brydon Carse arrived at the crease with his trunk injury and managed to secure a major win for Durham by playing a shot through gully and third slip.