The application of modifying nanoadditives in the technology of cement composites is currently a relevant and widely researched topic in global materials science. The purpose of this study was to investigate new nanoadditives—nanofibrils made from synthesized wollastonite (NF) and nanospheres from corundum (NS)—produced by LLC NPK Nanosystems (Rostov-on-Don, Russia) as a modifying additive. During the experimental investigations, the mechanical properties of cement pastes and concrete were examined. This included an analysis of the density, compressive and bending strength, as well as water absorption of concrete that had been modified with NF and NS additives. X-ray phase and microstructural analyses of concrete were performed. It was established that modification of cement composites with NF and NS additives had a beneficial effect on their properties, and the optimal amount for both types of additives was 0.3% by binder weight. The highest recorded enhancements in compressive and flexural strength of concrete with 0.3% NF were 7.22% and 7.04%, respectively, accompanied by a decrease in water absorption by 4.70%. When modifying concrete with 0.3% NS, the increases in compressive and flexural strength were 2.71% and 2.48%, and water absorption decreased by 1.96%. Modification of concrete with NF and NS additives did not have a significant effect on the change in concrete density, which was no more than 1%. Based on the results of phase analysis, it was established that concrete with NF and NS additives were characterized by the presence of five main phases: quartz, portlandite, calcite, larnite, and olivine-Ca. It was found that compositions with 0.3% NF and NS differed from the control composition by the presence of such a phase as olivine-Ca. Microstructural analysis confirmed the effectiveness of NF and NS additives. The microstructure of the modified concretes was distinguished by the extensive occurrence of clusters composed of calcium silicate hydrate zones. The conducted studies prove the possibility of using NF and NS as modifying nanoadditives in the technology of cement composites. The addition of nanofibrils from synthesized wollastonite is the most effective and promising and is recommended for use in real construction practice.