Darlo had recorded three wins and a draw before AFC Telford United spoilt their record with a 1-0 win at Blackwell Meadows.
Marcus Dinanga's goal for the visitors two minutes after half-time was enough to take home the points.
Darlington beat Salford City on the opening day and also drew at York City last Tuesday in front of a crowd of almost 4000, but their busy start caught up with them on the weekend.
"There was a lot of work gone in on Tuesday night, it was our toughest game that we had and I just think today we looked very tired," said manager Gray.
"We had enough chances off set plays and dead ball free-kick situations. Our players weren't up to their usual standards unfortunately.
"That team there [AFC Telford] is in a false position. They didn't really carve us open, they created probably one chance. It was a soft goal we conceded.
"We have to react now. We need the break and get them back on the training ground."