A three-goal burst in the first nine minutes killed off Darlington's hopes of a Worthington Cup shock at Valley Parade last night.
But Quakers, who made six changes to the team which drew with Macclesfield on Friday, left Valley Parade with their honour intact by playing some good football in the second half to pull back two goals.
Darlington fans were rewarded for their non-stop support by a superb penalty save by keeper Andy Collett from Benito Carbone near the end.
Darlington manager Gary Bennett admitted that he was very disappointed "Defensively we were all over the place. Some of the goals we conceded were schoolboy stuff," he said.
"We got off to a very bad start and gave them two silly goals early on, which you cannot do at this level. To be three goals down in nine minutes against a Premiership club ended up with us chasing the game. At that point I didn't think we would score at all.
"Even after half-time we conceded another bad goal to go 5-0 down and at that stage we were only playing for our pride.
"I came here thinking that we would still have a chance, but we certainly didn't after 15 minutes.
"I am very disappointed for our supporters who made the journey down because the team let the club down."
A goal down from the first leg, Quakers got off to the worst possible start.
After three minutes Ashley Ward chested down a throw-in from the left for Gareth Whalley to beat Collett from 12 yards.
Five minutes later Carbone chipped in a free-kick from the left for Dean Windass to head into the bottom corner of the net from six yards out.
The tie was virtually all over after nine minutes when Adam Reed, playing in the centre of defence, was woefully short with a back pass which let in Carbone, and he went on to beat Collett at the second attempt.
Bradford nearly got a fourth after 15 minutes when Carbone opened up the Darlington defence for Ward to shoot powerfully towards goal, but Collett saved well.
Then the keeper pulled off a good diving save at the foot of his post from a Windass free-kick from the edge of the box.
Bradford eased off slightly and Quakers managed to put together a couple of useful moves after Martin Gray came on for Mark Kilty.
They nearly pulled one back after 20 minutes when Mark Angel beat two men in a good run into the box from the right, but his low shot went just past the post.
Then Stuart Elliott tried a long- range shot which went just wide as they came more into the game, encouraged by their travelling support of around 700 fans.
But there was another body blow after 41 minutes courtesy of another bad mistake at the back. Carl Pepper tried to head a long ball from Peter Atherton back towards Collett, but Carbone nipped in again to beat Collett to the ball and roll it into the empty net.
Carbone nearly completed his hat-trick just on half-time with a bicycle kick from a Windass free-kick which flew just over the angle.
Quakers went further behind two minutes after half-time. Mark Bower squared the ball for Windass, who hit a swerving shot past Collett from 30 yards.
Bradford could have added further goals, but Reed twice put in good tackles on Gareth Grant as he threatened to break clear.
Quakers lifted the spirits of the travelling fans on the hour.
Elliott pounced on a loose ball just outside the area, raced into the box and even though Aidan Davison blocked his piledriver, the midfielder was first to react to volley into the empty net.
There was more delight for Quakers after 63 minutes when Richard Hodgson delivered a perfect cross with his left foot from the right for Angel to side-foot past Davison.
Grant made up for his misses by grabbing the sixth after 73 minutes. Ward rolled the ball across the Darlington box for Jamie Lawrence to have two shots charged down by Collett, but Grant was on hand to side-foot the loose ball home.
Bradford were then awarded a penalty after 75 minutes for a foul by Steven Walklate on Windass, but Collett pulled off a superb one-handed save down to his left.
Having used all their substitutes Quakers were reduced to ten men after 81 minutes when Pepper was stretchered off after a clash with Grant. Collett did not have any chance in the last minute when Carbone curled in a free kick for Gunnar Halle to beat the keeper with a diving header.
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