Generation of topological phases of matter with SU(3) symmetry in a condensed-matter setup is challenging due to the lack of an intrinsic threefold chirality of quasiparticles. We uncover two salient ingredients required to express a three-component lattice Hamiltonian in a SU(3) format with a nontrivial topological invariant. We find that all three SU(3) components must be entangled via a gauge field, with opposite chirality between any two components, and there must be band inversions between all three components in a given eigenstate. For spinless particles, we show that such chiral states can be obtained in a tripartite lattice with three inequivalent lattice sites in which the Bloch phase associated with the nearest-neighbor hopping acts as the k-space gauge field. The second and a more crucial criterion is that there must also be an odd-parity Zeeman-like term, i. e., sin(k) sz term, where sz is the third Pauli matrix defined in any two components of the SU(3) basis. Solving the electron-photon interaction term in a periodic potential with a modified tight-binding model, we show that such a term can be engineered with site-selective photon polarization. Such site-selective polarization can be obtained in multiple ways, such as using the Sisyphus cooling technique, polarizer plates, etc. With the k-resolved Berry curvature formalism, we delineate the relationship between the SU(3) chirality, band inversion, and k-space monopoles, governing the finite Chern number without breaking the time-reversal symmetry. The topological phase is affirmed by edge-state calculation, obeying the bulk-boundary correspondence.