Rob Hulse put Derby in front in the first half but Adam Hamill levelled before half-time with a stunning strike.
Andy Gray, a recent signing from Charlton, put the Tykes in front with a breakaway goal in the second half after Kris Commons appeared to be bundled over at the other end of the field.
Anderson De Silva added a third in stoppage time on the counter attack and although there was still time for Shaun Barker to score his first goal for the Rams, the damage had already been done.
Robins concluded following his first game in charge that the Tykes were suffering from a crisis of confidence and that became evident in the opening 20 minutes as Derby started to get into their stride.
Hulse should have opened his account for the season after 13 minutes when Jake Livermore played him in, but the Rams striker's scuffed shot was deflected wide.
Dean Moxey let fly with a venomous left-footed drive that flashed wide of the Barnsley goal.
But Nigel Clough's side were in front after 26 minutes when Lee Croft whipped in an inviting cross from the right and Hulse planted an unstoppable header past Tykes keeper David Preece.
Iain Hume had an opportunity to draw Barnsley level after 31 minutes but his free-kick from just outside the penalty area flashed wide.
An unmarked Hulse had a golden opportunity to double Derby's lead but somehow failed to connect with another precise cross from Croft.
And he was made to pay for that miss when Hamill produced a moment of magic to equalise nine minutes before the break.
The winger picked up the ball on the left corner of the penalty area and cut in before curling a sublime finish into the far corner of the net.
It took a superb last-ditch tackle from Bobby Hassell to deny Gary Teale what would have been a simple finish just after the break.
And Commons came close with a deflected free-kick soon after.
But it was the Tykes who went in front after 56 minutes with a goal that infuriated the home supporters.
Commons appeared to be bundled over as Derby attacked but referee Nigel Miller waved play on and a swift Barnsley break resulted in Emil Hallfredsson playing in Gray, who broke the offside trap and slotted the ball past Stephen Bywater.
Derby lost their way after Gray's goal and as they pushed forward for an equaliser Barnsley broke again and De Silva rounded Bywater and tapped the ball into an empty net in stoppage time.
Barker rose to head home a left-wing corner in the fifth minute of time added on but the Tykes held on for three precious points.