That win puts them straight through to the contest’s knockout stages.
Striker Kieran McGrath scored the only goal of the game four minutes before half-time with a free-kick from 25 yards out.
Aidan Denholm also hit the bar for the Northumbrians just ahead of the interval.
They go into that game top of the competition’s three-team south-east group with three points from one game, their Galashiels visitors being in second place with a single point and having already qualified too due to league champions Bonnyrigg Rose pulling out.
Their next fixture after that, away to East Kilbride in the league, has been put back to Tuesday, April 5, at 7.45pm.