We construct a minimal spinful tight-binding model on a square lattice, where a d-wave sublattice-staggered altermagnetism drives the quantum anomalous Hall effect. Here the exchange field is staggered between the two sublattices, where it takes opposite signs on A and B described by the Pauli matrix τ_z. The resulting insulating phases host tunable Chern numbers C=±1 and C=±2, controlled by the staggered exchange strength and the sublattice-staggered potential. We determine the complete phase diagram, identify valley-resolved band inversions at the X and Y points in the Brillouin zone, and demonstrate chiral edge states together with quantized two-terminal conductance plateaus. Our work provides a simple route to realizing the quantum anomalous Hall effect in compensated magnets via a d-wave sublattice-staggered altermagnetism.