Future generations of gravitational-wave detectors (GWD) are targeting an effective quantum noise reduction of 10 dB via the application of squeezed states of light. In the last joint observation run O3, the advanced large-scale GWDs LIGO and Virgo already used the squeezing technology, albeit with a moderate efficiency. Here, we report on the first successful 10 dB sensitivity enhancement of a shot-noise limited tabletop Michelson interferometer via squeezed light in the fundamental Gaussian laser mode, where we also implement the balanced homodyne detection scheme that is planned for the third GWD generation. In addition, we achieved a similarly strong quantum noise reduction when the interferometer was operated in higher-order Hermite-Gaussian modes, which are discussed for the GWD thermal noise mitigation. Our results are an important step toward the targeted quantum noise level in future GWDs and, moreover, represent significant progress in the application of nonclassical states in higher-order modes for interferometry, increased spatial resolution, and multichannel sensing.