Adam Boyes' scored his 11th goal of the season as Barrow opened up a four-point gap on Darlington.
Boyes' first-half header ensured the visitors recorded their second successive victory in the Blue Square Bet Premier.
Andy Cook could have doubled the Bluebirds' advantage three minutes after the interval, but his 20-yard effort flew over.
Darlington searched for a leveller, but James Walshaw fired over from 18 yards.
Adam Rundle and Liam Hatch went close for the hosts late on but the Quakers' two-match winning run came to an end.
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Darlington boss Mark Cooper tells BBC Tees:
"If you put that many balls in the box and you don't score you have to ask questions of the centre-forwards - and I'm asking questions of my centre-forwards.
"There were a ridiculous amount of balls that went in there, over 40, we've got an amazing habit of finding space where the ball isn't going to bounce instead of where it is going to bounce.
"If we had Jason Walker tonight then we'd have scored four or five, he's got another two tonight, but there we go."
Barrow joint-boss Dave Bayliss told BBC Radio Cumbria:
"We knew what we were facing tonight - it was going to be a real test of our mental strength for such a young side.
"Defensively, that was an outstanding team performance.
"We got a great goal. We had the most clear-cut chances during the match, and we always knew that their threat lay in throw-ins and set-pieces. We asked the players to put themselves in there and they've done that."