We show that periodic driving with near-infrared to visible light drives topological phase transitions in the photon-dressed band structure of transition-metal dichalcogenide heterobilayers. We apply the Floquet formalism to a light-coupled lowest-order k·p Hamiltonian, and obtain an effective four-band model that correctly captures the essential photon-dressed bands of type-II TMD heterobilayers in the vicinity of the first photon resonance. Crossings between the zero- and one-photon sectors effectively invert the bands, yielding topological phases with Chern numbers up to ± 2 and gaps of order 10 meV. Our results establish TMDs as prime candidates for engineering bands with higher Chern numbers, and exploring driven topological states in solid state media.