A 1-0 defeat to Austria in Bucharest means it is the Austrians who go through from Group C, while Ukraine must now wait on other results from the group stages to see if they can progress as one of the four best third-placed sides.
Christoph Baumgartner scored the only goal of the game for Austria on 21 minutes, with Andriy Shevchenko's side rarely threatening.
Zinchenko cut a frustrated figure, unable to get his passing game going as he played in several roles during the game.
There are various scenarios that could yet see Ukraine progress, but they are currently the fourth best of six sides hoping to go through with their chances on a knife-edge.
Austria now face Roberto Mancini's Italy at Wembley on Saturday in the Round of 16.
Elsewhere in Group C, the Netherlands beat North Macedonia 3-0 in Amsterdam to maintain their 100% record so far.
Nathan Ake was an unused sub.