We show that an unconventional superfluid triggered by spin-orbit coupling is realized for repulsively interacting quasi-one-dimensional fermions. A competition between spin-singlet and -triplet pairings occurs due to the breaking of inversion symmetry. We show that both superfluid correlations decay algebraically with the same exponent except for special coupling constants for which a dominant superfluid is controlled by the spin-orbit coupling. We also discuss a possible experiment to observe such phases with cold atoms.