Superconductivity frequently appears by doping compounds that show a collective phase transition. So far, however, this has not been observed in topological materials. Here we report the discovery of superconductivity induced by Ga doping in orthorhombic Re$_{3}$Ge$_{7}$, which undergoes a second-order metal-insulator-like transition at $sim$58 K and is predicted to have a nontrivial band topology. It is found that the substitution of Ga for Ge leads to hole doping in Re$_{3}$Ge$_{7-x}$Ga$_{x}$. As a consequence, the phase transition is gradually suppressed and disappears above $x$ = 0.2. At this $x$ value, superconductivity emerges and $T_{rm c}$ exhibits a dome-like doping dependence with a maximum value of 3.37 K at $x$ = 0.25. First-principles calculations suggest that the phase transition in Re$_{3}$Ge$_{7}$ is associated with an electronic instability driven by Fermi surface nesting and the nontrival band topology is preserved after Ga doping. Our results indicate that Ga-doped Re$_{3}$Ge$_{7}$ provides a rare opportunity to study the interplay between superconductivity and competing electronic states in a topologically nontrivial system.