The Euler Sombor index of a graph G is a recently introduced topological index, defined as \[ EU(G)~=~\sum_{uv\in{E(G)}}\sqrt{d(u)^2+d(v)^2+d(u)d(v)}, \] where \( d(u) \), \( d(v) \) are the degrees of the vertices \( u \), respectively \( v \) of \( G \). The purpose of this paper is to determine the first, second and third minimal and maximal unicyclic graphs of order \( n \) with respect to the Euler Sombor index for all \( n \geq 5 \).