The steering, control or switching of electron wavepackets by light is expected to pave the way towards the much wanted visualization of nanoplasmonic field dynamics or real-time probing of electron motion in solid nanostructures. Electron pulses generated by strong-field tunneling from sharp gold tips irradiated by few-cycle laser pulses and accelerated in the near field are believed to be suitable sources of such highly confined, coherent, ultrashort electron wavepackets. Here we show the effect of the carrier-envelope phase of the driving laser field on the generation and motion of such strong-field accelerated electrons. This is a crucial step forward towards the electron motion around nanostructures on nanometer-length and attosecond-time scale.