Context. The addition of a 28 m Cherenkov telescope (CT5) to the H.E.S.S. array extended the experiment’s sensitivityto lower energies. The lowest energy threshold is obtained using monoscopic analysis of data taken with CT5, providing access to gamma-ray energies below 100 GeV for small zenith angle observations. Such an extension of the instrument’s energy range is particularly beneficial for studies of active galactic nuclei with soft spectra, as expected for those at a redshift ≥0.5. The high-frequency peaked BL Lac objects PKS 2155−304 (z = 0.116) and PG 1553+113 (0.43 < z < 0.58) are among the brightest objects in the gamma-ray sky, both showing clear signatures of gamma-ray absorption at E > 100 GeV interpreted as being due to interactions with the extragalactic background light (EBL). Aims. The aims of this work are twofold: to demonstrate the monoscopic analysis of CT5 data with a low energy threshold, and to obtain accurate measurements of the spectral energy distributions (SED) of PKS 2155−304 and PG 1553+113 near their SED peaks at energies ≈100 GeV. Methods. Multiple observational campaigns of PKS 2155−304 and PG 1553+113 were conducted during 2013 and 2014 using the full H.E.S.S. II instrument (CT1–5). A monoscopic analysis of the data taken with the new CT5 telescope was developed along with an investigation into the systematic uncertainties on the spectral parameters which are derived from this analysis. Results. Using the data from CT5, the energy spectra of PKS 2155−304 and PG 1553+113 were reconstructed down to conservative threshold energies of 80 GeV for PKS 2155−304, which transits near zenith, and 110 GeV for the more northern PG 1553+113. The measured spectra, well fitted in both cases by a log-parabola spectral model (with a 5.0σ statistical preference for non-zero curvature for PKS 2155−304 and 4.5σ for PG 1553+113), were found consistent with spectra derived from contemporaneous Fermi-LAT data, indicating a sharp break in the observed spectra of both sources at E ≈ 100 GeV. When corrected for EBL absorption, the intrinsic H.E.S.S. II mono and Fermi-LAT spectrum of PKS 2155−304 was found to show significant curvature. For PG 1553+113, however, no significant detection of curvature in the intrinsic spectrum could be found within statistical and systematic uncertainties.