Win by a nose to win by the slightest amount of difference. (Informal. As in a horse race where one horse wins with only its nose ahead of the horse that comes in second.)
If somebody wins by a nose, they only just beat the others.
In other words win by a whisker. Just barely succeed
I ran the fastest race I could, but I only won by a nose.
Sally won the race, but she only won by a nose.
My horse won but only by a nose. In fact it was a very exciting finish.
The candidate won by a nose so they had to count the votes again.