Anchorage failure of a suction anchor is more likely to occur in low-strength muddy clay. This paper focuses on the failure behaviors of suction anchors and muddy clay stress responses. The centrifugal model test was used to study the loading processes of suction anchors with various pulling angles. Firstly, the multi-stage developing process of anchoring force was analyzed according to the test results. Numerical modeling was used to validate the test results. The displacement of the suction anchor and muddy clay soil were analyzed using the numerical results. Then, the numerical and testing results were compared to analyze the horizontal soil pressure responses around the suction anchors. It was found that the change in loading direction affected the distribution and development of soil stress. The horizontal soil resistance played a crucial role in improving the bearing capacity. The soil stress variation and anchor displacement revealed that the suction anchors exhibited multi-attitude coupling movement during the inclined pulling. The vertical pulling suction anchor showed shear–slip failure behaviors, while the inclined pulling suction anchors showed compression–shear–slip coupling failure behaviors. The results of this study provide insight into the interaction mechanism between suction anchors and muddy clay, serving as a reference for the design and application of suction anchors.