We demonstrate experimentally the manipulation of supercurrent in Al-AlOx-Ti Josephson tunnel junctions by injecting quasiparticles in a Ti island from two additional tunnel-coupled Al superconducting reservoirs. Both supercurrent enhancement and quenching with respect to equilibrium are achieved. We demonstrate cooling of the Ti line by quasiparticle injection from the normal state deep into the superconducting phase. A model based on heat transport and the nonmonotonic current-voltage characteristic of a Josephson junction satisfactorily accounts for our findings.