As a special kind of tweezers, the acoustic-vortex (AV) beam with a bent or inclined trajectory shows its great significance in contactless manipulations, especially for objects behind obstacles. By introducing additional phase delays to the traditional single-side ring-array of planar transducers, directional off-axis acoustic-vortex (OA-AV) beams passing through a preassigned point are constructed. Numerical simulations and experimental measurements of the axial and cross-sectional profiles for OA-AV beams of different orders with various radial offsets are conducted for transducers of various radiation patterns. An approximately linear trajectory of OA-AV beams with the slope determined by the preassigned point is demonstrated. The axial distribution of OA-AV beams formed by the main-lobes and side-lobes with corresponding zero-pressure nodes is decided by the directivity of the sources, while the vortex radius can be adjusted by the topological charge. By combining with the technologies of ultrasonic imaging, acoustic treatment, particle assembly, and targeted drug delivery, the precise positioning and flexible regulation of tilted OA-AV beams can be applied to accurately route acoustic packets along a predictable trajectory in three dimensions, suggesting more promising potentials of obstacle-avoidant object manipulation in biomedical applications.