narrow victory; razor-thin win
Idiomatic expression likening a close victory to walking on thin ice; emphasizes the precariousness of the win.
The election result was a narrow victory.
He advanced to the final with a razor-thin win.
辛勝 is a more formal, Sino-Japanese term for a narrow victory, while 薄氷の勝利 is an idiomatic expression with a vivid image of precariousness.
僅差 refers to a narrow margin in general, not necessarily a victory; 薄氷の勝利 specifically describes a win that was barely achieved.
From 薄氷 (thin ice) + の + 勝利 (victory), literally 'victory on thin ice', conveying the idea of a win that could easily have been a loss.