Resummation of the soft-gluon radiative corrections for the quark-vector boson vertex is performed within the path-integral (world-line) approach. The leading-order expression for the vacuum-averaged Wilson integral for an arbitrary gauge field is found in n-dimensional space-time. The cusp anomalous dimension of the color non-singlet Sudakov form factor of on-mass-shell quark is calculated in an arbitrary covariant gauge in the one-loop order, and the leading double-logarithmic asymptotical behavior is obtained from the corresponding evolution equation.