Broad interest in quantum spin liquid (QSL) phases was triggered by the notion that they can be viewed as insulating phases with preexisting electron-pairs, such that upon light doping they might automatically yield superconductivity. Yet despite intense efforts, definitive evidence is lacking. We address the problem of a lightly doped QSL through a large-scale density-matrix renormalization group study of the $t$-$J$ model on the triangular lattice with a small but non-zero concentration of doped holes. The ground state is consistent with a Luther-Emery liquid with power-law superconducting and charge-density-wave correlations associated with partially-filled charge stripes. In particular, the superconducting correlations are dominant on both four-leg and six-leg cylinders at all hole doping concentrations. Our results provide direct evidences that doping a QSL can naturally lead to robust superconductivity.