First half goals by Nathan Cartman and David Syers gave Quakers their sixth away league win of the season at York Street, and their first at Boston in 27 years.
Quakers were well in control at half time with goals by Cartman and Syers, and they missed several chances to kill the game off.
Boston then threw everything at Quakers, but after they pulled one back, Quakers defended well with keeper Ed Wilczynski outstanding.
Boston had a free kick on 4 minutes on the left hand side about 30 yards out, but Adam Chapman fired the ball high over the bar.
Boston had the first corner on 8 minutes that was swung in by Chapman, but Quakers easily got the ball away.
Boston had a chance on 10 minutes when Hawley sent Hilliard away, but he rolled the ball wide from 25 yards.
Quakers took the lead after 14 minutes, when a long throw by Gary Brown was flicked on by Kevin Burgess, and the ball ran to the far post where Nathan Cartman tapped the ball home for his fifth league goal of the season.
Quakers nearly went 2-0 up on 26 minutes when David Ferguson curled a corner over from the right for David Syers to head firmly towards goal, but Boston keeper Ross Durrant pulled off a good save.
Boston tried to get back into it and a right wing cross dropped for Hawley, but his low shot was blocked.
David Ferguson was fouled on the left, he swung the free kick into the free kick where it was only cleared as far as Syers twelve yards out, and he volleyed the ball first time left footed into the bottom corner of the net for his sixth goal of the season.
Boston United 0 Darlington 2
Thompson had a good run down the right in the opening minute of the second half and got into the box, but his pull back was blocked.
Thompson then popped up on the left and took on two defenders, fired towards goal and Durrant saved at full stretch.
Boston had a chance on 57 minutes when a throw in dropped for Hawley in the box, his shot was charged down and sub Colby Bishop lifted the rebound over the bar..
But the home side pulled a goal back on 59 minutes when Hawley fired past Wilczynski from twelve yards after the ball was knocked down to him.
And it was nearly 2-2 when Simmons got through, but Wilczynski saved at his feet.
Thompson was unlucky at the other end with a right foot shot that Durrant saved at the first attempt.
Wilczynski then pulled off an excellent one handed save to stop a free kick from Simmons going in the bottom corner.
Boston had another chance when Hawley set up Rollins but wilczynski blocked with his feet, then Simmons curled the follow up over the bar.
But Quakers held on strongly after that, without really allowing Boston any clear sight on goal, although there was a penalty shout late on when Simmons went down but the ref waved play on.
Boston United 1 Darlington 2
Wilczynski, Falkingham, Ferguson, Turnbull (sub Galbraith 80), Burgess, Brown, Syers, Scott, Beck, Cartman (sub Hardy 70), Thompson (sub Gillies 90+5)
Subs not used: Bell, Blackford
Durrant, Gatter, Robinson, Batchelor, Yeomans, Clarke, Chapman, Hilliard, Simmons, Hawley, Rollins
Subs: Thomas, Gordon, Fitzpatrick, Chippendale, Bishop