A second-half strike from Canterbury forward Anna Bennett was enough to give Great Britain's Olympic hockey hopes a lift in Auckland.
They remain in fourth spot in Pool A, with the top two teams qualifying automatically and four others from the groups facing play-off matches.
Bennett scrambled home the winner after 53 minutes to give Britain their first victory in their third match.
Eimear Cregan put Ireland ahead against Germany, but the match finished 1-1.
British coach Tricia Heberle was far from impressed with her team's display against Ukraine. We did our best to keep Ukraine in the tournament
"We talked about being patient and passing the ball. We needed to be more composed and consistent for 50-60 minutes before wearing them down.
"We did our best to keep Ukraine in the tournament. We will forget this game and have to lift ourselves against Ireland."
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