Robust and accurate acceleration tracking remains a challenge in many fields. For geophysics and economic geology, precise gravity mapping requires onboard sensors combined with accurate positioning and navigation systems. Cold atom–based quantum inertial sensors can potentially provide these high-precision instruments. However, current scalar instruments require precise alignment with vector quantities. Here, we present the first hybrid three-axis accelerometer exploiting the quantum advantage to measure the full acceleration vector by combining three orthogonal atom interferometer measurements with a classical navigation-grade accelerometer triad. Its ultralow bias permits tracking the acceleration vector over long time scales, yielding a 50-fold improvement in stability (6 × 10 −8 g ) over our classical accelerometers. We record the acceleration vector at a high data rate (1 kHz), with absolute magnitude accuracy below 10 μ g , and pointing accuracy of 4 μrad. This paves the way toward future strapdown applications with quantum sensors and highlights their potential as future inertial navigation units.