The structure of quasicrystalline Al$_{65}$Co$_{20}$Cu$_{15}$ and Al$_{72}$Co$_{18}$Ni$_{10}$ alloys as well as composites on their base fabricated by furnace impregnation is investigated in this work. The methods of metallography, X-ray diffraction, scanning electron microscopy and energy-dispersive X-ray spectrometry are applied. In the Al$_{65}$Co$_{20}$Cu$_{15}$ alloy quasicrystalline decagonal phase co-exists with crystalline Al$_4$(Co, Cu)$_3$ and Al$_3$(Cu, Co)$_2$ phases, and in the Al$_{72}$Co$_{18}$Ni$_{10}$ alloy—with Al$_9$(Co$_{1-x}$Ni$_x$)$_2$ and Al$_9$(Ni$_{1-x}$Co$_x$)$_2$ phases. Volume fraction of quasicrystalline phase in the alloys ranges from 60 to 65% vol. Using original method of automatized structural analysis, distribution curves of absorption factors are plotted to calculate entropy of the phases. During impregnation of the Al$_{65}$Co$_{20}$Cu$_{15}$ or Al$_{72}$Co$_{18}$Ni$_{10}$ filler granules by brass L62 binder, the molten binder dissolves crystalline phases of the fillers penetrating up to the centre of the granules. At that, quasicrystalline phase of fillers dissolves at much lesser rate. In the structure of the composites reinforced with Al$_{65}$Co$_{20}$Cu$_{15}$ filler, a volume fraction of the quasicrystalline phase exceeds by 15% vol. this phase content in the composites reinforced with Al$_{72}$Co$_{18}$Ni$_{10}$ filler. To explain the difference in the dissolution rates of the filler’s phases during infiltration, a quantity and an average geometric vibration frequency of oscillators in their structure are calculated. The model conceptions of theory of strongly anisotropic crystals are used. The quasicrystalline phase of the fillers is characterized by the largest amount and the lowest vibration frequency of oscillators. The calculations also confirm that the quasicrystalline phase of the Al$_{65}$Co$_{20}$Cu$_{15}$ filler has higher resistance to molten binder influence during impregnation compared with that of the quasicrystalline phase of the Al$_{72}$Co$_{18}$Ni$_{10}$ filler. At the same time, crystalline phases of the Al$_{65}$Co$_{20}$Cu$_{15}$ alloy dissolve with the higher rate, which assures strong adhesion between the filler and the solidified binder. The Al$_{65}$Co$_{20}$Cu$_{15}$ filler is recommended for fabrication of the composites designated for operation under dry friction and effect of acidic solutions.