  The strong coupling, $\alpha_s$, governs perturbative Quantum Chromodynamics
(QCD) and is one of the free parameters of the Standard Model. We introduce a
new method that allows a precise extraction of $\alpha_s(m_Z)$ from
dimensionless ratios of roots of moments of the charm-quark vector correlator.
The ratios we use in our analysis have a rather weak logarithmic quark-mass
dependence, starting at $\mathcal{O}(\alpha_s^2)$, and can be obtained from
experimental data with good precision, since they benefit from positive
correlations among the individual experimentally determined moments. We perform
a careful and conservative error analysis with special emphasis on
uncertainties related to the truncation of perturbation theory, treating the
renormalization scales such as to ensure order-by-order convergence. Our final
result, with expressions at $\mathcal{O}(\alpha_s^3)$, is
$\alpha_s(m_Z)=0.1168\pm 0.0019$.
