The crossword solution for the term 'BARELY BEATS, WITH 'OUT'' is EDGES. This phrase is often used to describe a situation where one person or team barely wins over another. It implies that the winner was only able to win by a small margin and that the other person or team was very close to winning. The addition of the word 'OUT' implies that the winner was able to do something that the other person or team was not able to do, allowing them to edge OUT the victory.