Due to the absence of periodic length scale, electronic states and their topological properties in quasicrystals have been barely understood. Here, we focus on one-dimensional quasicrystals and reveal that their electronic critical states are topologically robust. Based on tiling space cohomology, we exemplify the case of one-dimensional aperiodic tilings especially Fibonacci quasicrystals and prove the existence of topological critical states at zero energy. Furthermore, we also show exotic electronic transmittance behavior near such topological critical states. Within the perturbative regime, we discuss the lack of translational symmetries and that the presence of topological critical states lead to an unconventional scaling behavior in transmittance. Considering both analytic analysis and numerics, electronic transmittance is computed in cases where the system is placed in air or is connected by semi-infinite periodic leads. Finally, we also discuss a generalization of our analysis to other quasicrystals. Our findings open a new class of topological quantum states which solely exist in quasicrystals due to exotic tiling patterns in the absence of periodic length scale, and their anomalous electronic transport properties applicable to many experiments.