Morten Olsen's men had already secured top spot in Group One and a place in South Africa next summer, but Buzsaky's first-half goal resulted in their first defeat of the campaign.
The hosts dominated the early stages and should have taken the lead after nine minutes when Jon Dahl Tomasson picked out Nicklas Bendtner with a perfect pass. The Arsenal striker scuffed his shot but Thomas Enevoldsen latched onto the ball before firing over the crossbar from close range.
Jacob Poulsen had the next opening for the Danes but directed his effort past the post, while Anders Moller Christensen was prevented from scoring by goalkeeper Gabor Babos with his 21st-minute header from Enevoldsen's free-kick. Babos pulled off another good save three minutes later, with Michael Jakobsen getting a foot to a free-kick from Bendtner.
The chances continued to go begging for the Scandinavian side, with Jakobsen firing wide following a good lay-off from debutant Enevoldsen.
They were made to pay ten minutes before the interval when a poor clearance from Christensen fell into the path of Buzsaky, who lifted the ball over goalkeeper Thomas Sorensen and into the net.
Denmark tried to get back on level terms in the second half but Babos was up to the challenge, punching a fierce Daniel Agger drive from 35 yards away for a corner.
Sorensen was also proving his worth at the other end of the pitch and prevented Hungary from doubling their lead after 70 minutes when he tipped Sandor Torghelle's close-range strike onto the crossbar.
With the clock ticking down, Michael Silberbauer hoped to reward Denmark's persistence with an equaliser, but a frustrating evening was capped when Babos produced a brilliant save to stop the winger's 25-yard drive from finding the top right corner.