Set 170 to win after Ben Cotton’s maiden five-wicket haul helped bowl Sussex out for 222 in their second-innings, Northants slipped from 53 for one to 82 for four.
But Curran’s solid half-century - his second in his five first-class matches since making his debut in August - and 51 not out from Adam Rossington saw their side home.
Sussex collapsed from 114 for two after lunch and their season ended with a third successive away defeat - they have won only once away from home in 2018.
They had a chance to improve that record after moving into a good position just after lunch as Harry Finch and Luke Wells rescued Sussex from five for two, trailing by 48, to raise a century stand for the third wicket as two batsmen finally got to grips with conditions after 20 wickets fell on day one.
But like day one, poor strokes outside off saw both Finch and Wells dismissed within three overs of each other - Finch for 63 caught behind off Ben Sanderson and Wells caught at first slip for 50 to become Cotton’s first victim.
Ben Brown was then strangled down the leg side before a superb inswinger from Cotton saw Delray Rawlings record a pair in his only appearance this season.
Michael Burgess chipped a catch to point and when Cotton ran a full-straight delivery into Danny Briggs’ stumps, he completed his haul.
Five for 48 was a timely return, especially as Cotton is playing for a contract for next season.
There was a brief Sussex fightback as Chris Jordan and Ollie Robinson added 41 for the eighth wicket but Robinson edged a Nathan Buck lifter behind and Jordan ran himself out for 47 with only No. 11 Abidine Sakande for company.
It left Northants with a modest target but a challenge after a season of consistently failing in crucial moments of matches.
Nerves were settled as Curran and Ricardo Vasconcelos - who clipped Jordan over square for a boundary to avoid a pair - added 53 for the opening wicket.
But Vasconcelos followed a Robinson delivery angled across him and edged to slip for 19 and it sparked a wobble.
Charlie Thurston was caught at the wicket for four from a beauty from Robinson that went away from the right-hander, Alex Wakely edged to second slip and Luke Procter was pinned lbw for one.
Procter’s wicket was Robinson’s 74th of the season - the most wickets in a season by a Sussex seamer since James Kirtley took 75 in the 2001 division two title triumph.
It was too much to ask, though, for Robinson to deliver victory as Rossington joined Curran to ease any tension and their sprightly unbroken stand of 90 brought a positive end to Northants’ season of struggle.