In this paper, we introduced two novel subclasses of bi-univalent functions, ${\mathcal{M}}_{\mathrm{\Sigma }}(\alpha ,\mathcal{B}(x,\xi ))$ and ${\mathcal{H}}_{\mathrm{\Sigma }}(\alpha ,\mu ,\mathcal{B}(x,\xi ))$, utilizing Lucas-Balancing polynomials. Within these function classes, we established bounds for the Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$, addressing the Fekete-Szegö functional problems specific to functions within these new subclasses. Moreover, we illustrated how our primary findings could lead to various new outcomes through parameter specialization.