Topologically close-packed (TCP) phases exhibit great promise as high-temperature structural materials, however, their notorious brittleness at room temperature limits their applications. The dislocation motion mechanisms in these complex alloys, especially on non-basal/\{111\} planes, remain poorly understood. We perform atomistic simulations to elucidate the deformation mechanisms of TCP phases and complement with experimental observations from nanomechanical testing at room temperature. Dislocation glide and cross-slip mechanisms among newly identified \{11n\} slip planes in cubic Laves phases and associated minimum energy paths are determined. Additionally, we unveil a new non-basal slip mechanism, namely the formation of (1-105) stacking fault by partial dislocation glide with a Burgers vector of 0.07[-5502], in μ-phases. This comprehensive exploration of dislocation dynamics not only contributes to a fundamental understanding of TCP phases but also paves the way for tailored design strategies to enhance their mechanical properties at room temperature.