Sidelobe elimination is essential in practical acoustic focusing applications for reducing energy leakage and signal interference in non-target regions. Acoustic metasurfaces (AMs) provide an effective platform for sidelobe suppression due to their abilities to precisely manipulate the amplitude and phase of incident waves. However, most existing AMs are either passive or rely on complex feeding networks for simultaneous amplitude and phase modulation (APM), which severely limits their flexibility in implementing optimized amplitude-weighting schemes for effective suppression. Here, we propose a sidelobe-free acoustic focusing strategy based on a space-time-coding (STC) metasurface, enabling efficient sidelobe elimination at different focal positions. By exploiting spatiotemporal acoustic impedance modulation, the STC metasurface achieves simultaneous and precise APM, with a continuously tunable amplitude range of 0-1 and a full 0-2π phase coverage. Based on the amplitude-weighted sidelobe suppression theory, the time-varying parameters of the metasurface unit cells are dynamically engineered to synthesize a Gaussian amplitude distribution and the phase gradient required for acoustic focusing across the metasurface aperture. As a result, acoustic energy can be efficiently concentrated at the desired focal location, with sidelobe components significantly suppressed. Specifically, we demonstrate high-quality sidelobe-free focusing at different positions, such as (-2.5λ, 3λ) and (1.5λ, 4λ), where the average peak sidelobe ratio (PSLR) is reduced by approximately 81.0% and 74.6%, respectively. This work provides a versatile and reconfigurable approach for controllable acoustic sidelobe suppression and facilitates precise acoustic focusing in applications such as ultrasonic therapy and nondestructive testing.