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We investigate numerically the time evolution of wave packets incident on one-dimensional semi-infinite lattices with mosaic modulated random on-site potentials, which are characterized by the integer-valued modulation period $κ$ and the disorder strength $W$. For Gaussian wave packets with the central energy $E_0$ and a small spectral width, we perform extensive numerical calculations of the disorder-averaged time-dependent reflectance, $\langle R(t)\rangle$, for various values of $E_0$, $κ$, and $W$. We find that the long-time behavior of $\langle R(t)\rangle$ obeys a power law of the form $t^{-γ}$ in all cases. In the presence of the mosaic modulation, $γ$ is equal to 2 for almost all values of $E_0$, implying the onset of the Anderson localization, while at a finite number of discrete values of $E_0$ dependent on $κ$, $γ$ approaches 3/2, implying the onset of the classical diffusion. This phenomenon is independent of the disorder strength and arises in a quasi-resonant manner such that $γ$ varies rapidly from 3/2 to 2 in a narrow energy range as $E_0$ varies away from the quasi-resonance values. We deduce a simple analytical formula for the quasi-resonance energies and provide an explanation of the delocalization phenomenon based on the interplay between randomness and band structure and the node structure of the wave functions. We explore the nature of the states at the quasi-resonance energies using a finite-size scaling analysis of the average participation ratio and find that the states are neither extended nor exponentially localized, but ciritical states..