Barrow boss Pete Wild admitted he would have been happy with a point at Bradford before George Ray popped up late on to secure a 2-1 win.
Defender Ray sealed a hard-earned victory which spoiled Graham Alexander’s first game in charge of the Bantams.
The Bluebirds are now up to sixth in League Two thanks to a seven-match unbeaten run.
Wild said: “I’m 100 per cent delighted with that result.
“We knew the thoughts about new manager bounce get thrown around but we had a pattern and a game-plan we’d spoken about before the game.
“Then to score so early was fantastic. I think that stunned Bradford a bit while we were solid, on the front foot and were winning every header.
“We were more than matching them. We got penned in a bit more in the second half but that happens coming to places like this.
“We had to sit in for the ride but again we were winning tackles, winning headers.
“As the game went on I would have settled for a point but to then go on and win the game playing with a 5-4-1 formation like that was just brilliant.
“These lads just have that never-say-die attitude and this club as a whole is at one.
“We’re definitely all pulling in the right direction.”
It was an entertaining first half, which began disastrously for new boss Alexander when Dom Telford fired Barrow ahead after only 30 seconds.
However, the Bantams rallied and by the interval were level thanks to Tyler Smith’s close-range finish just past the half-hour mark.
The Bantams had the better of the second period but Ray won the game for Barrow with 13 minutes to go as he pounced on a loose ball from eight yards.
It was a frustrating opening game for Alexander, who took the reins at Valley Parade on Monday.
He said: “We couldn’t have started the game any worse.
“You’re then looking at how the players react to that blow and how we can affect the rest of the game from there.
“It took us a bit of time to get going, and there’s reasons for that, but we did find it hard to break them down in that first 25 minutes.
“We got the goal we deserved though and could have had another when the keeper made a great save.
“We made a few tactical adjustments at half-time and I did think we at least saw the fruits of that in the second half.
“We were the dominant team and should have won.
“Perhaps we were looking for the perfect goal at times later in the game, so we have got to hit the target more when we get the chance to.
“I can’t fault the effort and the commitment of the players, though.
“They certainly bought into what I want to do in that second half before we got done by a sucker punch.”