edge out

verb

1. To defeat in a contest or a game by a narrow margin of victory.

   “Barcelona edged out in the penalties scoring 3-2.”

   “Crawley missed two penalties but still edged out League Two Torquay to become the first non-league side to reach the FA Cup fifth round for 17 years.”

2. To gradually exclude; to push someone or something further and further into the margins until they/it is entirely outside of a space.

   “Since the Second World War and the rise of the middle-class "clone gay” in the US and a similar move away from homosexual effeminacy in Britain—often rooted in working class culture—gender nonconforming or effeminate gay males have been edged out of mainstream understandings of what it means to be gay.”