|Edit Eördögh had 11 points for Hungary|
And that relief is due to the most unlikely of sources, centre Eva Seres, who made a three-pointer with just four seconds left to lift her side to victory.
Staring a 56-54 deficit, Seres took aim and buried her only shot from long range on the night.
Ratgeber has fretted over the play in the low post where he believes Hungary are at a disadvantage, so it is no small irony that Seres would save the team on opening day.
The Hungarians, getting their customary 16-point performance from Andrea Karolyi, were unable to shake off the Ukraine in an ugly contest where poor shooting was the norm.
Ratgeber's troops led by five points entering the final frame.
Hungary might not have been involved in such a nail-biter had they not struggled from the free throw line where they made a woeful 11 of 20 in the game.
His other centre who has been hobbled with a foot problem, Krisztina Papp, only played for one minute and 30 seconds and missed both of her shots.
Edid Eordogh was effective, making five of her six shots from the floor, including one-of-one from three-point range, but Ratgeber will be looking for much more efficiency on offence even though he has said in the buildup to the tournament that defence will have to carry his side.
Hungary were just 20 of 58 from the field for the game, including six of 19 from three-point land.
The Ukraine had four people in double figures, including Ole Zherzherunova who had a team-high 16 points, but the Ukrainians were awful shooting the ball as they sank just 16 of their 56 shots from the field.
Hungary host Belgium on Saturday while the Ukraine now face the unenviable task of a clash with Russia, the favourites in this competition.