Keaton Jennings and Usman Arshad continued where they left off the previous day, adding runs to the lead at a good rate. The pair got the lead to 280 and put on 106 together before Jennings found Brown off Nash to depart for 127.
Arshad passed 50 for the first time in his Championship career and was in fine touch with the bat, getting to 83 from 88 balls (12x4s and 3x6s) before Ashar Zaidi claimed his wicket and Joyce the catch.
Chris Rushworth and Graham Onions added a further 14 runs for the final wicket before Rushworth was bowled by Ashar Zaidi and Durham were all out for 458 with a lead of 294.
Needing 295 to win, Sussex approached their run chase with purpose and in style as they punished the Durham attack and deducted runs quickly from the scoreboard.
Openers Nash and Wells added a record 163-run stand before Graham Onions claimed the breakthrough, taking his 70th wicket for the season courtesy of a sensational catch to Collingwood.
Needing 132, Nash continued to attack the Durham bowlers and made his way to a century before Scott Borthwick got his wicket shortly after he raised his bat, making 108 before he found the hands of Richardson.
It was then left to Ed Joyce and Matthew Prior with the target under 50 runs away however Prior made just the 6 before Will Smith took a wicket in his first over and Borthwick the catch.
In a strange turn of events, Phil Mustard came into the attack for his bowl of his First Class career and on his fifth ball claimed the wicket of England limited overs representative Luke Wright.
Despite the loss of those two wickets, Sussex were already over the line however needing just 4 more runs and Ed Joyce (65no) ensured his side finished the season on a high to claim third position.
Despite a loss in the final match, Durham showed their fighting spirit in this fixtures and nothing can take away from the season they had taking out the LV=County Championship with 10 victories in what will be recorded in history as a successful season for the side.