We present initial results from the low-latitude Galactic plane region of the High Time Resolution Universe pulsar survey conducted at the Parkes 64-m radio telescope. We discuss the computational challenges arising from the processing of the terabyte-sized survey data. Two new radio interference mitigation techniques are introduced, as well as a partially coherent segmented acceleration search algorithm which aims to increase our chances of discovering highly relativistic short-orbit binary systems, covering a parameter space including potential pulsar-black hole binaries. We show that under a constant acceleration approximation, a ratio of data length over orbital period of ≈0.1 results in the highest effectiveness for this search algorithm. From the 50 per cent of data processed thus far, we have redetected 435 previously known pulsars and discovered a further 60 pulsars, two of which are fast-spinning pulsars with periods less than 30 ms. PSR J1101-6424 is a millisecond pulsar whose heavy white dwarf (WD) companion and short spin period of 5.1 ms indicate a rare example of full-recycling via Case A Roche lobe overflow. PSR J1757-27 appears to be an isolated recycled pulsar with a relatively long spin period of 17 ms. In addition, PSR J1244-6359 is a mildly recycled binary system with a heavy WD companion, PSR J1755-25 has a significant orbital eccentricity of 0.09 and PSR J1759-24 is likely to be a long-orbit eclipsing binary with orbital period of the order of tens of years. Comparison of our newly discovered pulsar sample to the known population suggests that they belong to an older population. Furthermore, we demonstrate that our current pulsar detection yield is as expected from population synthesis.