Chasing 261 to win, Durham began on 15-1 and nightwatchman Mark Wood (66) hit his highest first-class score as he put on 116 with Keaton Jennings (61).
Notts took two quick wickets before Scott Borthwick's 51 took Durham close.
Michael Richardson (51 not out) ensured Durham completed a three-day win, leaving Notts still without a victory.
Nottinghamshire have now drawn three and lost one of their opening four matches in the Championship.
Durham reduced their victory target to 154 runs when they reached lunch on 107-1.
Notts gave themselves hope as Luke Wood had Jennings caught by Steven Mullaney at gully, while Harry Gurney got one to move sharply to dismiss Mark Wood six overs later.
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Wood, who did not feature in England's recent tour of West Indies, spent 114 balls at the crease, hitting a six and nine fours, passing his previous best of 58 not out at Trent Bridge two years ago.
Durham then eased to their victory target, thanks largely to a stand of 86 in 14.4 overs between Borthwick and Richardson.
Borthwick hit eight fours in his 65-ball knock for a fourth half-century of the season, before being trapped lbw by Samit Patel.
Durham suffered a dramatic collapse in their last match against Middlesex when, chasing 259, they were bowled out for only 71.
Calum MacLeod (21 not out) made sure there would be no such problems and struck Patel for a six and a four in one over, before hitting the winning run off the same bowler.
Durham batsman Keaton Jennings: "For Mark Wood to get a career-best on that pitch was phenomenal.
"He's a tremendous competitor and he looked calm and controlled. He took the pressure off me.
"We tried to play every ball on its merit, knowing we needed to get through the first hour. To get through the morning unscathed was a real bonus."
Nottinghamshire director of cricket Mick Newell: "There wasn't much swing but there was still a bit of movement off the seam.
"We beat the bat a bit but from a slightly shorter length than is needed here. It's not a difficult ground to bowl on.
"If you have a first-innings lead of 83 here and then lose you have probably bowled pretty poorly."
BBC Newcastle's Martin Emmerson: "I am not really sure why it is but Durham continue to hold a spell over Notts - this was their fifth win in six games.
"Having been bowled out for just 163 on Monday in their second batting horror show of the last week, you wouldn't have given them much hope of chasing down 261, the highest total in the game.
"Eighteen wickets fell on Monday but by lunchtime none had gone today.
"Credit to Keaton Jennings and Mark Wood for their stand of 116 runs, while it was nice to see half-centuries for Scott Borthwick and Michael Richardson as they helped restore some timely confidence."

Staff at Inverness-based Lifescan Scotland were informed of the development by management on Tuesday.
It comes after parent company Johnson and Johnson announced it was reviewing its strategic options.
LifeScan makes products for the treatment of diabetes and currently employs 1,100 people in the Highland capital.
The products include meters and software to let people self-monitor their blood glucose levels.
Johnson and Johnson reported slower than expected sales to the financial markets on Tuesday and said they were looking to sell or find partners or other options for Lifescan.
Last summer Highlands and Islands Enterprise (HIE) invested Â£1.8m in Lifescan to support 600 existing jobs.

Wembley was being rebuilt, so the match was played at Millennium Stadium in Cardiff and was most notable for the stunning - and some would say unexpected - strike from Ray Parlour that sealed victory for Arsene Wenger's team.
Thierry Henry, Frank Lampard, Dennis Bergkamp, Marcel Desailly - there were plenty of big names on the pitch back then, but what are they up to now?
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Durham beat Nottinghamshire by six wickets to secure their third Championship win of 2015 and continue their good start to the season.
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Up to 80 jobs are under threat at one of Scotland's largest life sciences companies.
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Arsenal and Chelsea have only featured in one previous FA Cup final in 2002 - when the Gunners defeated their London rivals 2-0.