For a simple graph \( G \) with vertex degrees \( d_G(\cdot) \), the elliptic Sombor index (ESO) is defined as \( \text{ESO}(G) = \sum_{xy \in E(G)} (d_G(x) + d_G(y)) \sqrt{d_G^2(x) + d_G^2(y)} \). We completely solve the extremal problem for this index among connected tricyclic graphs of order \( n \), characterizing both the unique graph achieving the minimum ESO value and all graphs attaining the maximum value. These results provide sharp bounds for the elliptic Sombor index in this important graph class and reveal new extremal structural properties not observed in other degree-based topological indices.