When the pair came together Durham had a lead of 156, with around 30 overs still remaining in the contest. Carse’s eventual dismissal, for 31, meant that the target had jumped to 202 from 17 overs.
Notts gave it a go before agreeing to call off the chase on 82 for one, with only 7.4 overs left.
Peter Trego twice cleared the ropes in hitting 34 from 22 balls and Duckett was unbeaten on 36 from 28.
Earlier, the visitors were eventually bowled out for 329 in their second innings, with Samit Patel claiming four for 80.
The home county were held up for more than an hour by Chris Rushworth at the start of the day, with the nightwatchman eventually falling for 25.
Jones, playing in only his fifth first-class fixture, defied Notts for 248 minutes in making his highest first-class score.
Following a duck in the first innings Jones played a chanceless knock, hitting eight fours in his 224-ball stay before nicking Jake Ball behind soon after the second new ball had been taken.
Trego struck twice after lunch, getting David Bedingham lbw for 29 and then forcing Stuart Poynter to misjudge a leave that ricocheted back onto his stumps.
Carter had Gareth Harte caught at short leg by Haseeb Hameed for 34 before Coughlin nicked Patel behind.
Ned Eckersley only made eight but took 75 minutes out of the game, eventually falling to Patel after facing 73 deliveries.
Carse and Raine then played their punchy cameos to put any thoughts of a victory charge for Notts back on ice.
Requiring around 12 runs an over from the start Trego started as if it could be achieved, slog-sweeping Rushworth over the ropes before launching Raine even further.
The first six overs produced 66 runs, but the fall of Trego’s wicket and the sight of nine Durham fielders hugging the boundary prompted the early finish.
Nottinghamshire take 16 points, but their wait for a first red ball victory since June 2018 goes on.