Durham 337 (Stoneman 113, Hastings 51, Hatchett 5-113) and 359 for 6 dec (Jennings 103, Borthwick 95) beat Sussex 205 (Wright 66, Stokes 7-67) and 182 (Tredwell 50*, Hastings 4-47) by 309 runs
Durham needed just 17 balls to take the final Sussex wicket on Wednesday morning to secure a 309-run victory that saw them move to fourth in the Division One standings.
Resuming on 41, James Tredwell edged Chris Rushworth's second ball of the morning through the slips for four before driving the next to the cover boundary.
A single took him to 50 but he was then left stranded when Lewis Hatchett edged Rushworth to third slip to wrap up a convincing Durham victory - their second of the season in the four-day format.
Durham have the chance to secure their third win when they face Division One leaders Yorkshire at Headingley on July 7-10.
Victorious skipper Paul Collingwood said: "I'm delighted with the performance. I wasn't sure what to do when I won the toss and the first hour was tricky. But after that we steadily built the pressure and having people score runs down the order is crucial.
"Ben Stokes changed the game in their first innings and his bowling was superb throughout. It's good for Durham to have him when we expected him to be in the Test team.
"He is the type of character who wants to prove people wrong and he has proved his fitness with long spells in this game.''
Sussex, who have slipped to third bottom in the standings, are at home to bottom club Northamptonshire in their next match, starting on July 6.
Coach Mark Robinson said: "We are in a relegation battle but there are so few teams in the division you are nearly always battling neat the top or bottom. We conceded too many runs in the first innings here after having them 148 for five.''