A method for exciting high-order optical vortex modes in an annular-core fiber is proposed. The method involves controlling the tilt angle and the offset distance between the tapered and lensed single mode fiber (SMF) and the annular-core fiber. Numerical simulation and experimental verification results show that the excitation of the high quality second-order optical vortex mode can be realized by optimizing the tilt angle and the offset distance to approximately 8° and 2 μm, respectively. This tilted offset excitation method based on a tapered and lensed SMF can improve the coupling efficiency because of the focusing effect of the tapered and lensed SMF. Compared with excitation by a standard SMF, an enhancement in the coupling efficiency of approximately 13% is found by using the tapered and lensed SMF. The excited high-order optical vortex modes show considerable potential for high-resolution microscopy, optical micromanipulation, optical sensing, etc.