We present a technique for the synthesis of ultrashort laser pulses with approximately one cycle (FWHM) of temporal duration. These pulses are characterized by a certain degree of chirp. We show that these pulses produce both an enhancement of the high-order harmonic generation (HHG) cutoff and a noticeable increase of the yield, when interacting with an atomic system. Additionally, the asymmetric nature of the driven pulses plays an important role in the efficiency and cutoff extension of the high-order harmonics generated. Starting from the HHG spectra, we demonstrate it is possible to retrieve isolated attosecond pulses by spectral filtering. The analysis and interpretation of the different characteristics present in the HHG driven by this kind of pulse was carried out invoking classic arguments. Furthermore, a more complete description and validation of the HHG properties is performed by a quantum analysis, based on the integration of the time-dependent Schrödinger equation in full dimensionality.