Presented are the results of calculations suggesting that the quasi-one dimensional organic superconductors (TMTSF)₂X may show a substantial increase in their superconducting and spin density-wave ordering temperatures when the Fermi level is raised through application of an electrostatic gating voltage.  A rich behavior is observed, strongly dependent on the form of the superconducting order parameter, as the Fermi level approaches the van Hove singularity at k_a = 0.  Included are predictions for the behavior of these materials under zero and moderate applied pressure.  It is found that T_SDW as high as 50 K and superconducting T_c as high as 20 K may be achieved at optimal gate voltages of approximately 100 mV.