Biased diffusive transport of Brownian particles through irregularly shaped, narrow confining quasi-one-dimensional structures is investigated. The complexity of the higher dimensional diffusive dynamics is reduced by means of the so-called Fick-Jacobs approximation, yielding an effective one-dimensional stochastic dynamics. Accordingly, the elimination of transverse, equilibrated degrees of freedom stemming from geometrical confinements and/or bottlenecks causes entropic potential barriers that the particles have to overcome when moving forward noisily. The applicability and the validity of the reduced kinetic description are tested by comparing the approximation with Brownian dynamics simulations in full configuration space. This non-equilibrium transport in such quasi-one-dimensional irregular structures implies, for moderate-to-strong bias, a characteristic violation of the Sutherland-Einstein fluctuation-dissipation relation.