Austria were able to celebrate their achievement, however, the result leaves Ukraine sweating on what seems a slim chance they'll be one of the four best third-placed teams and also earn their place in the knockout stages.
Baumgartner turned in David Alaba's corner midway through the first half as Austria profited from a lacklustre screen from Ukraine, who fought to find fluidity when it seemingly mattered most in the tournament.
The Netherlands, who beat North Macedonia 3-0 in their final match, top the group with a complete haul of nine points from their three games, while Austria are next on six points and Ukraine have three, and a drawback goal-difference.
Austria may not be celebrating for long though as they confront in-form Italy at the past 16 in London on Saturday, coming against Roberto Mancini's team that are unbeaten in their past 30 internationals and that have not conceded in 11 games.
Austria made the early running and went ahead on 21 minutes via a simply-worked goal. Alaba's corner delivery reached Baumgartner, who'd moved ahead of his marker and prodded the ball to the internet.
Ukraine mostly wasted what ownership they had with misplaced passes but made a rare opening just before the half-hour mark when Mykola Shaparenko's shot was beaten away by Austria goalkeeper Daniel Bachmann and Andriy Yarmolenko narrowly failed to catch onto the rally.
Ukraine were smarter in the start of the second half and showed more ambition to get forward and place Austria under stress.
But a lack of accuracy with their passing continued to hamper the side and the only time that they seemed threatening was from occasional set-piece opportunities.
In the closing stages, as the moments ticked off, Ukraine never looked like getting the target that would have meant that they finished over the Austrians.