We present a detection of the enhancement in the number densities of background galaxies induced from lensing magnification and use it to test the Sunyaev–Zel'dovich effect (SZE-) inferred masses in a sample of 19 galaxy clusters with median redshift z ≃ 0.42 selected from the South Pole Telescope SPT-SZ survey. These clusters are observed by the Megacam on the Magellan Clay Telescope though gri filters. Two background galaxy populations are selected for this study through their photometric colours; they have median redshifts z median ≃ 0.9 (low-z background) and z median ≃ 1.8 (high-z background). Stacking these populations, we detect the magnification bias effect at 3.3σ and 1.3σ for the low- and high-z backgrounds, respectively. We fit Navarro, Frenk and White models simultaneously to all observed magnification bias profiles to estimate the multiplicative factor η that describes the ratio of the weak lensing mass to the mass inferred from the SZE observable-mass relation. We further quantify systematic uncertainties in η resulting from the photometric noise and bias, the cluster galaxy contamination and the estimations of the background properties. The resulting η for the combined background populations with 1σ uncertainties is 0.83 ± 0.24(stat) ± 0.074(sys), indicating good consistency between the lensing and the SZE-inferred masses. We use our best-fitting η to predict the weak lensing shear profiles and compare these predictions with observations, showing agreement between the magnification and shear mass constraints. This work demonstrates the promise of using the magnification as a complementary method to estimate cluster masses in large surveys.