Scanning tunneling microscopy (STM) enables the bottom-up fabrication of tailored spin systems on a surface that are engineered with atomic precision. When combining STM with electron spin resonance (ESR), these single atomic and molecular spins can be controlled quantum-coherently and utilized as electron-spin qubits. Here we demonstrate universal quantum control of such a spin qubit on a surface by employing coherent control along two distinct directions, achieved with two consecutive radio-frequency (RF) pulses with a well-defined phase difference. We first show transformations of each Cartesian component of a Bloch vector on the quantization axis, followed by ESR-STM detection. Then we demonstrate the ability to generate an arbitrary superposition state of a single spin qubit by using two-axis control schemes, in which experimental data show excellent agreement with simulations. Finally, we present an implementation of two-axis control in dynamical decoupling. Our work extends the scope of STM-based pulsed ESR, highlighting the potential of this technique for quantum gate operations of electron-spin qubits on a surface.