Edge Out Meaning

(idiomatic) To win in a contest or a game by a narrow margin of victory.

Example: 2011 January 29, Chris Bevan, “Torquay 0 - 1 Crawley Town”[1], BBC:
  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.