Robins was particularly furious at two penalty incidents which went against his side.
He firstly saw his team denied a spot-kick when Haines adjudged that a handball offence had taken place right on the edge of the box - and then the referee awarded the visitors a penalty of their own for a foul by Ryan Shotton on the fleet-footed Nicky Maynard midway through the second half.
Although Jamie McAllister duly missed from the spot, the follow-up was turned in by Marvin Elliott and Bristol City were on their way.
And to make matters worse for Barnsley they were reduced to ten men after Shotton was shown the red card.
Robins said: "I'm not going to comment fully on the referee and his performance because I'd find myself in deep trouble.
"But when you've played the game you can clearly see that the Bristol lad had over-run the ball and wasn't pulled back. But once he's given the penalty he's got to send Ryan off and suddenly we're up against it."
The outcome could have been so different had Barnsley got their noses in front following a string of early chances.
Iceland international Emil Hallfredsson came close to a brilliant opener in the 12th minute but his 20-yard thunderbolt was superbly finger-tipped onto the underside of the bar by City keeper Dean Gerken.
Jamal Campbell-Ryce and Adam Hammill then both spurned decent openings before a mistake by veteran defender Darren Moore enabled Maynard to burst clear and drill the visitors in front in the 26th minute with his ninth goal of the season.
City always looked threatening going forward, with the pace of Maynard and Danny Haynes constantly troubling the Barnsley rearguard and, three minutes before half-time, Haynes bamboozled full-back Carl Dickinson and pulled the ball back to Evander Sno who finished smartly left footed.
Robins threw on Malta international Daniel Bogdanovic at the start of the second half and he threatened to haul Barnsley back into the contest when his 62nd-minute shot forced a mistake from Gerken, who allowed the ball to slip through his grasp and through his legs into the net.
Barnsley were now playing their best football but were denied their spot-kick appeals and, to make matters worse, referee Haines then adjudged that Shotton had been guilty of pulling a shirt and awarded City a 73rd-minute spot-kick.
McAllister saw his effort brilliantly saved by Luke Steele but Elliott followed up to score.
A great stoppage time goal by Hammill proved to be little more than a consolation for the home team.