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In laser-wakefield acceleration, an ultra-intense laser pulse is focused into an underdense plasma to accelerate electrons to relativistic velocities. In most cases, the pulses consist of multiple optical cycles and the interaction is well described in the framework of the ponderomotive force where only the envelope of the laser has to be considered. But when using single-cycle pulses, the ponderomotive approximation breaks down, and the actual waveform of the laser has to be taken into account. In this paper, we use near-single-cycle laser pulses to drive a laser-wakefield accelerator. We observe variations of the electron beam pointing on the order of 10 mrad in the polarization direction, as well as 30% variations of the beam charge, locked to the value of the controlled laser carrier-envelope phase, in both nitrogen and helium plasma. Those findings are explained through particle-in-cell simulations indicating that low-emittance, ultrashort electron bunches are periodically injected off-axis by the transversally oscillating bubble associated with the slipping carrier-envelope phase. [ABSTRACT FROM AUTHOR]