Use 'edge out' to describe a victory that's achieved by a very small margin. He managed to edge out the other candidate by just a few votes.
Pro Tip 2/3
Progressive Ousting
'Edge out' can mean slowly displacing someone from their position, often in a subtle way. With new policies, she slowly edged her rival out of influence.
Pro Tip 3/3
Not Aggressive
The phrase 'edge out' is not typically used to describe an aggressive or forceful action. He didn't force his competitors away, but edged them out through consistent performance.