Callum Thorp took 6-17 as Scotland were bowled out for 161 in a rain-hit game at The Grange on Sunday.
Durham were reduced to four for two, and they were still in trouble at 32 for three, but Gary Pratt's 43 and Dale Benkenstein's 27 not out ensured a five-wicket win which puts them level with Lancashire at the top of the North Conference.
Meanwhile, Ben Phillips kept his nerve as Northamptonshire recorded a dramatic one-run victory over Derbyshire at Wantage Road.
Bilal Shafayat's 55 was the top score in the hosts' total of 257, with Graham Wagg picking up a four-wicket haul.
Derbyshire got within a run of that total when Stephen Stubbings, who had scored 110, was run out and Phillips then dismissed Jake Needham.
Anurag Singh struck an unbeaten 78 as Nottinghamshire defeated Warwickshire by seven wickets at Edgbaston.
Rookie opener Navdeep Poonia had given the hosts a promising start with 59 from 81 balls in his first Bears appearance, but they could only post 186.
In the South Conference, Andy Flower and James Foster guided Essex to a four-wicket win over Kent at Canterbury.
Chris Peploe picked up four wickets as Middlesex defeated Somerset by three runs in Bath.
Cheltenham&Gloucester Trophy -- North Conference: Northamptonshire 257 (B Shafayat 55, D Sales 50, G Wagg 4-59) beat Derbyshire 256 (S Stubbings 110) by one run.
At The Citylets Grange: Scotland 161 (C Thorp 6-17) lost to Durham 99-5 by 5 wkts (D/L method).
At Edgbaston: Warwickshire 186 (N Poonia 59) lost to Nottinghamshire 189-3 (A Singh 78 no) by 7 wkts.
South Conference at Bristol: Gloucestershire 339-8 (I Harvey 108, D Brown 63 no) beat Surrey 337 (J Benning 189 no) by two runs.
At Canterbury: Kent 281-6 (R Key 89, M Walker 52 no) lost to Essex 282-6 (M Pettini 80, A Flower 61) by 4 wkts.
At Bath: Middlesex 215-9 (O Shah 63, E Morgan 50 no) beat Somerset 212-9 (J Francis 51, C Peploe 4-41) by three runs.