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Neuromorphic systems consisting of artificial neurons and memristive synapses could provide a much better performance and a significantly more energy-efficient approach to the implementation of different types of neural network algorithms than traditional hardware with the Von-Neumann architecture. However, the memristive weight adjustment in the formal neuromorphic networks by the standard back-propagation techniques suffers from poor device-to-device reproducibility. One of the most promising approaches to overcome this problem is to use local learning rules for spiking neuromorphic architectures which potentially could be adaptive to the variability issue mentioned above. Different kinds of local rules for learning spiking systems are mostly realized on a bio-inspired spike-time-dependent plasticity (STDP) mechanism, which is an improved type of classical Hebbian learning. Whereas the STDP-like mechanism has already been shown to emerge naturally in memristive devices, the demonstration of its self-adaptive learning property, potentially overcoming the variability problem, is more challenging and has yet to be reported. Here we experimentally demonstrate an STDP-based learning protocol that ensures self-adaptation of the memristor resistive states, after only a very few spikes, and makes the plasticity sensitive only to the input signal configuration, but neither to the initial state of the devices nor their device-to-device variability. Then, it is shown that the self-adaptive learning of a spiking neuron with memristive weights on rate-coded patterns could also be realized with hardware-based STDP rules. The experiments have been carried out with nanocomposite-based (Co40Fe40B20) (LiNbO3−y)100− memristive structures, but their results are believed to be applicable to a wide range of memristive devices. All the experimental data were supported and extended by numerical simulations. There is a hope that the obtained results pave the way for building up reliable spiking neuromorphic systems composed of partially unreliable analog memristive elements, with a more complex architecture and the capability of unsupervised learning.