Context. High-energy (HE) and very high-energy (VHE) gamma-ray observations from the Galactic center (GC) detected extended emission correlated with the morphology of the central molecular zone (CMZ). Emission in both bands is expected to be produced by hadronic interaction between cosmic rays (CRs) and ambient gas. Aims. We examine if our three previously proposed scenarios for the CR sources and dynamics, which are consistent with the VHE gamma-ray data (1–100 TeV), also match the HE gamma-ray observations (10–300 GeV). Additionally, we analyze the effect of the isotropic Galactic CR “sea” inside the CMZ. Methods. We generated synthetic gamma-ray maps considering a simplified isotropic diffusion, but more realistic dynamics with two diffusion zones (in and out of the CMZ) and polar advection, for mono-energetic particles of 3 TeV. Additionally, we considered two gas distributions for the CMZ (with and without an inner cavity), and CR populations injected from the clusters of young massive stars (the Arches Cluster, the Quintuplet Cluster, and the nuclear star cluster), plus the supernova Sgr A East. Results. Only the combination of more realistic CR dynamics, the CMZ with an inner cavity, CR injection from all proposed sources, and a CR sea similar to that observed in the Solar System reproduced the current HE and VHE gamma-ray detection from the CMZ and was consistent with the observed gamma-rays from Sagittarius A*. Conclusions. The HE and VHE gamma-rays observations of the GC can be reproduced by a unified model for the CRs.