In practical engineering, submarine cables inevitably cross oblique slip faults, and currently, the local buckling instability mode of buried submarine cables under the action of oblique slip faults with both horizontal and vertical movements is not clear. A three-dimensional numerical model of submarine cable oblique slip fault was established based on ABAQUS finite element software, and corresponding research was carried out on the local buckling instability of various structural layers of submarine cable under oblique slip fault. The influence of fault dip angle, soil elastic modulus, and submarine cable burial depth on the critical buckling displacement under oblique slip fault was analyzed. The results indicate that under the action of oblique slip faults, stress concentration occurs in the armored steel wire of submarine cables located at the fault plane and diffuses to both sides with increasing displacement; The deformation of each structural layer is consistent and shows an "S" shape. The outer layer, armored steel wire, and copper conductor undergo buckling instability in order from the outside to the inside; When the dip angle of the fault is 90 °, the submarine cable structure has the strongest ability to withstand the action of oblique sliding faults; Local buckling and instability of submarine cables are more likely to occur in hard seabed soil; Appropriately increasing the burial depth of submarine cables can enhance their ability to resist the effects of oblique slip faults. The obtained results can provide reference for the protection of submarine cables crossing faults and the determination of local buckling in engineering.