We study pairing states in an largely imbalanced two-component Fermi gas loaded in an anisotropic two-dimensional optical lattice, where the spin up and spin down fermions filled to the $s$- and $p_x$-orbital bands, respectively. We show that due to the relative inversion of band structures of the $s$ and $p_x$ orbitals, the system favors pairing between two fermions on the same side of the Brillouin zone, leading to a large stable regime for states with finite center-of-mass momentum, i.e., the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. In particular, when the two Fermi surfaces are close in momentum space, a nesting effect stabilizes a special kind of $π$-FFLO phase with spatial modulation of $π$ along the easily tunneled $x$-direction. We map out the zero temperature phase diagrams within mean-field approach for various aspect ratio within the two-dimensional plane, and calculate the Berezinskii-Kosterlitz-Thouless (BKT) transition temperatures $T_{\rm BKT}$ for different phases. ; Comment: 6 pages, 4 figures