Off-axis three-mirror anastigmatic (TMA) telescopes with designed decenters and tilts, have many misalignment degrees of freedom and strong coupling between each misalignment degree of freedom. Therefore, it is difficult to establish misalignment equations only using A222 and A131 in nodal aberration theory (NAT). In addition, for off-axis TMA optical systems with designed decenters and tilts, the robustness of the existing fifth-order NAT misalignment calculation algorithm based on high-order Zernike coefficients and boresight errors decreases, so it is difficult to realize its engineering application. To solve the issue of insufficient practicality of the existing misalignment algorithm based on fifth-order NAT, a third-order NAT calculation algorithm based on quadratic aberration field decenter vectors is derived and established. Two concepts of inherent aberration field decenter vector and misalignment aberration field decenter vector are proposed. Taking an off-axis TMA optical system with 6-m focal length as the research object, simulations, and alignment verification experiments were carried out. Compared with the existing fifth-order NAT misalignment algorithm, the results show that when measurement noise is not considered, the two methods can both obtain convergent calculation results, and the average RMS wavefront errors (WFE) of the optical system are both corrected to be below 0.0574 waves. When different levels of measurement noise are introduced, the robustness of the fifth-order NAT misalignment algorithm decreases, and there are even cases where the optical system completely fails to be corrected. However, the algorithm based on quadratic aberration field decenter vectors shows better robustness. Under different levels of measurement noise, this algorithm could correct the average RMS WFE of the optical system to around 0.0574 waves.