We present a statistical weak-lensing magnification analysis on an optically selected sample of 3029 CAMIRA galaxy clusters with richness N > 15 in the Subaru Hyper Suprime-Cam (HSC) survey. The CAMIRA sample spans a wide redshift range of 0.2 z < 1.1. We use two distinct populations of color-selected, flux-limited background galaxies, namely the low-z and high-z samples at mean redshifts of ≈ 1.1 and ≈ 1.4, 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 CAMIRA sample is detected at a significance level of 8.29σ , which is dominated by the high-z background. We forward-model the observed magnification data to constrain the richness-to-mass (N-M) relation for the CAMIRA sample. In this work, we can only constrain the normalization of the N-M relation by employing informative priors on the mass and redshift trends, and on the intrinsic scatter at fixed mass. The resulting scaling relation is N ∝ M 500 0.91±0.14 (1 + z) −0.45±0.75 , with a characteristic richness of N = (19.63 ± 3.16) and intrinsic log-normal scatter of 0.14 ± 0.07 at M 500 = 10 14 h −1 M . With the derived N-M relation, we provide magnification-calibrated mass estimates of individual CAMIRA clusters, with the typical uncertainty of ≈ 38% and ≈ 30% at richness≈ 20 and ≈ 40, respectively. We further compare our magnification-inferred N-M relation with those from the shear-based results in the literature, finding good agreement.