Determined to get off to a good start, both nations had chances in the opening 15 minutes, with Daniel Lukacs directing Andras Schafer's knockdown onto the bar, before Danylo Sikan dragged a shot just wide at the other end.
Hungary captain Dominik Szoboszlai then drove his team forward and cut back for Bendeguz Bolla, who somehow volleyed over from a few yards out.
In an end-to-end contest, Viktor Tsyhankov then also fired off-target as his strike flew past the post.
Either side could have taken the lead before half-time, but it was Ukraine who struck first blood through Sikan, who headed home Georgiy Sudakov’s perfect cross.
Hungary did their best to respond early in the second half, and a well-worked corner found its way to Aron Csongvai, who looped his header onto the roof of the net.
Schafer then created another chance with a knockdown to substitute Milan Toth, who volleyed straight into Dmytro Riznyk’s grateful arms.
If the hosts were to equalise, it looked like Schafer would be involved, and he soon threatened again with an effort from distance that was also gathered by Riznyk.
Substitute Donat Barany also headed wide in a promising position, as the hosts continued to dominate into the latter stages.
However, Ukraine threatened on the break and could have put the game to bed through substitute Igor Krasnopir, who rounded Balazs Toth, but the goalkeeper did well to reach for the ball and keep it out.
The visitors ultimately stood firm despite Oleksandr Nazarenko receiving a second yellow card and subsequent red for a late challenge in stoppage time.
As such, they record their sixth win in nine matches (L3), with a visit to Georgia their next challenge under Maldera, who becomes the first foreign Ukraine manager to win a competitive international.
Hungary lose just their second in nine internationals (W5, D2), leaving Marco Rossi with much to ponder heading into their next fixture against Northern Ireland.
Flashscore Man of the Match: Oleg Ocheretko (Ukraine)