Sonoporation employs ultrasound accompanied by microbubble (MB) cavitation to induce the reversible disruption of cell membranes and has been exploited as a promising intracellular macromolecular delivery strategy. Due to the damage to cells resulting from strong cavitation, it is difficult to balance efficient delivery and high survival rates. In this paper, a traveling surface acoustic wave (TSAW) device, consisting of a TSAW chip and a polydimethylsiloxane (PDMS) channel, was designed to explore single-cell sonoporation using targeted microbubbles (TMBs) in a non-cavitation regime. A TSAW was applied to precisely manipulate the movement of the TMBs attached to MDA-MB-231 cells, leading to sonoporation at a single-cell level. The impact of input voltage and the number of TMBs on cell sonoporation was investigated. In addition, the physical mechanisms of bubble cavitation or the acoustic radiation force (ARF) for cell sonoporation were analyzed. The TMBs excited by an ARF directly propelled cell membrane deformation, leading to reversible perforation in the cell membrane. When two TMBs adhered to the cell surface and the input voltage was 350 mVpp, the cell sonoporation efficiency went up to 83%.