The striker took his tally to 12 for the season with an early strike and a second-half penalty to rock Forest.
Barnsley made a sensational start when they went in front with their first attack of note after 12 minutes.
Nathan Doyle's free-kick into the area was half-cleared and Bogdanovic conjured up a looping header that deceived keeper Lee Camp who back-pedalled furiously as the ball dropped over him.
The home side might have doubled their advantage three minutes later when Adam Hammill hit an audacious shot that was aimed at the far post but this time Camp dived full stretch to turn it round the post.
But Forest bounced back and saw two golden opportunities wasted. Veteran striker Dele Adebola menaced as Forest looked to hit back but guided a shot wide after Barnsley were caught napping at the back and then slammed another glorious chance wide as keeper Luke Steele was beaten.
And when Robert Earnshaw found a gap on the break he hesitated and his shot was blocked.
Another chance fell Barnsley's way early in the second half when Forest failed to clear from a corner but Bogdanovic screwed his shot wide.
Emil Hallfredsson carved his way through Forest with a surging run and slid through a smart pass but Jon Macken was inches short of connecting.
Forest, who had still to produce a shot on target, saw two more go astray when Chris Cohen and Radoslaw Majewski's efforts were wayward.
But Macken was guilty of wasting an opportunity at the other end when he burst through a gap but saw his shot blocked when Bogdanovic was free in the middle.
The second goal came after 65 minutes when Filipe Teixeira won a penalty after he was cut down in the area as he ran in on goal. Bogdanovic made no mistake from the spot.
Substitute Dexter Blackstock came on and pulled one back 12 minutes from time when he followed up his own blocked header and squeezed the ball over the line but it was too little too late for the visitors.