Nash, who runs the team along with Spartans first-team manager Michael Nelson in association with Tyne Met college, watched his side cruise into the first round qualifying.
Joseph Hands struck twice in the first-half with Jay Errington also getting on the scoresheet as Darlington were reduced to 10 men after just eight minutes.
Jamie Norvell also claimed a brace before substitute Matty Nesbitt rounded off the scoring from the penalty spot.
For many of the young Spartans, it was their first competitive game since 2020.
“I feel a huge sense of pride,” said Nash, who is also first-team manager at Northern League Division One side North Shields.
“Most of the lads that played had a largely redundant season last year due to Covid.
“They’ve been chomping at the bit to play competitively for over 12 months now.
“I was absolutely delighted with their attitudes and the way they’ve gone about it.”
Spartans host Billingham Town in the next round on the week commencing September 13, who are full of confidence after beating Scarborough Athletic 10-1.
Nash added: “Billingham are churning out young players who are getting into their first-team and it will be another test for our lads”