Three of Rangers' goals came in the first half - two of them in the opening 15 minutes - to leave Barnsley shell-shocked.
The Tykes briefly threatened an amazing comeback but, in the end, were well beaten.
Mikele Leigertwood gave QPR an early lead after storming forward from right-back and firing home from just outside the penalty area after linking well with Ben Watson and Martin Rowlands.
Watson then threaded through a perfect pass for Buzsaky, who steered the ball into the far corner of the net for his first goal since returning from a long-term knee injury.
On-loan Wigan midfielder Watson was outstanding in the first half and produced another clever pass for Wayne Routledge, only for the winger to shoot inches wide.
But it was Buzsaky who produced the game's champagne moment, collecting the ball on the left flank and sending an unstoppable 30-yarder into the top corner to put Rangers three up.
Barnsley were heading for a thrashing but hit back with two goals in the opening 11 minutes of the second half.
First Stephen Foster took advantage of some lazy QPR defending to head in Iain Hume's free-kick.
Then Barnsley were awarded a penalty after Damion Stewart brought down Adam Hammill.
Andy Gray saw his spot-kick saved by Radek Cerny but pounced on the rebound to score his third Championship goal of the season.
Rangers' nerves were eased when Watson netted after another lovely piece of play by Buzsaky on the left flank.
Buzsaky's cross was headed into Watson's path by Rowan Vine and after fumbling the former Crystal Palace man's close-range shot, keeper David Preece was unable to prevent the ball trickling over the line.
And Rangers netted a fifth when Alejandro Faurlin's 80th-minute corner was headed towards goal by Kaspars Gorkss and on-loan Arsenal forward Jay Simpson applied the finishing touch.