We present a statistical weak-lensing magnification analysis on an optically selected sample of 3029 texttt{CAMIRA} galaxy clusters with richness $N>15$ at redshift $0.2leq z <1.1$ in the Subaru Hyper Suprime-Cam (HSC) survey. We use two distinct populations of color-selected, flux-limited background galaxies, namely the low-$z$ and high-$z$ samples at mean redshifts of $approx1.1$ and $approx1.4$, respectively, from which to measure the weak-lensing magnification signal by accounting for cluster contamination as well as masking effects. Our magnification bias measurements are found to be uncontaminated according to validation tests against the null-test samples for which the net magnification bias is expected to vanish. The magnification bias for the full texttt{CAMIRA} sample is detected at a significance level of $9.51sigma$, which is dominated by the high-$z$ background. We forward-model the observed magnification data to constrain the normalization of the richness-to-mass ($N$--$M$) relation for the texttt{CAMIRA} sample with informative priors on other parameters. The resulting scaling relation is $Npropto {M_{500}}^{0.92pm0.13} (1 + z)^{-0.48pm0.69}$, with a characteristic richness of $N=left(17.72pm2.60right)$ and intrinsic log-normal scatter of $0.15pm0.07$ at $M_{500} = 10^{14}h^{-1}M_{odot}$. With the derived $N$--$M$ relation, we provide magnification-calibrated mass estimates of individual texttt{CAMIRA} clusters, with the typical uncertainty of $approx39%$ and $approx32%$ at richness$approx20$ and $approx40$, respectively. We further compare our magnification-inferred $N$--$M$ relation with those from the shear-based results in the literature, finding good agreement.
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