Abstract Assessing the stability of quasi-periodic (QP) response is crucial, as the bifurcation of QP response is usually accompanied by a stability reversal. The largest Lyapunov exponent (LLE), as an important indicator for chaotic motion, can also be used for the stability analysis of QP response. The precise location of a stability reversal, however, is tough to achieve as a poor convergence rate would be usually encountered when solving the LLE. Herein a straightforward and precise approach is suggested to identify the critical point where a stability reversal happens. Our approach can provide the LLE straightforwardly via numerical integration made to an explicit differential equation, and the corresponding covariant Lyapunov vector is simultaneously obtained. The major finding consists in the phase transition of the covariant Lyapunov vector, which can happen much early before the LLE reaches a relatively stable value. More importantly, the phase transition may occur even when there is no evident convergent value for the LLE. Consequently, the phase transition can serve as a strong indicator to locate the stability reversal qualitatively yet precisely. Numerical examples are provided to verify the effectiveness and wide applicability the presented approach.