  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 $z\simeq0.42$ selected from the South Pole
Telescope SPT-SZ survey. Two background galaxy populations are selected for
this study through their photometric colours; they have median redshifts
${z}_{\mathrm{median}}\simeq0.9$ (low-$z$ background) and
${z}_{\mathrm{median}}\simeq1.8$ (high-$z$ background). Stacking these
populations, we detect the magnification bias effect at $3.3\sigma$ and
$1.3\sigma$ for the low- and high-$z$ backgrounds, respectively. We fit NFW
models simultaneously to all observed magnification bias profiles to estimate
the multiplicative factor $\eta$ 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 $\eta$ resulting from the photometric
noise and bias, the cluster galaxy contamination and the estimations of the
background properties. The resulting $\eta$ for the combined background
populations with $1\sigma$ uncertainties is
$0.83\pm0.24\mathrm{(stat)}\pm0.074\mathrm{(sys)}$, indicating good consistency
between the lensing and the SZE-inferred masses. We use our best-fit $\eta$ 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.
