This study deals with a spatially homogeneous locally rotationally symmetric Bianchi type-I dark energy cosmological model containing a one-dimensional cosmic string fluid source. Einstein’s field equations are solved by using a relation between the metric potentials and hybrid expansion law of the average scale factor. We discuss the accelerated expansion of our model through the equation of state (EOS; ωde) and deceleration parameter (q). We observe that in the evolution of our model, the EOS parameter starts from matter-dominated phase ωde > −1/3 and ultimately attains a constant value in the quintessence region (−1 < ωde < −1/3). The EOS parameter of the model never crosses the phantom divide line (ωde = −1). These facts are consistent with recent observations. We also discuss some other physical parameters.