Spatiotemporal optical vortex (STOV) pulses can carry transverse orbital angular momentum (OAM) that is perpendicular to the direction of pulse propagation. For a STOV pulse, its spatiotemporal profile can be significantly distorted due to unbalanced dispersive and diffractive phases. This may limit its use in many research applications, where a long interaction length and a tight confinement of the pulse are needed. The first demonstration of STOV pulse propagation through a few-mode optical fiber is presented. Both numerical and experimental analysis on the propagation of STOV pulse through a commercially available SMF-28 standard telecommunication fiber is performed. The spatiotemporal phase feature of the pulse can be well kept after the pulse propagates a few-meter length through the fiber even with bending. Further propagation of the pulse will result in a breakup of its spatiotemporal spiral phase structure due to an excessive amount of modal group delay dispersion. The stable and robust transmission of transverse photonic OAM through optical fiber may open new opportunities for transverse photonic OAM studies in telecommunications, OAM lasers, and nonlinear fiber-optical research.