It is desirable to avoid the error and noise sensitivity of second-order tensor data leading to the disadvantages of low positioning accuracy with the Euler inversion method of magnetic object single-point positioning. For single-point positioning with only first-order tensor data, a method based on tensor derivative invariant relations is proposed. For the analysis of the magnetic dipole source tensor invariant and eigenvalue, two tensor derivative invariant relations were derived. The angle between the magnetic moment and position vector is constant and related to the tensor eigenvalue. The eigenvector of the absolute-minimum eigenvalue is perpendicular to the magnetic moment and position vector, and the eigenvectors of the remaining eigenvalues are coplanar with them. Thus, four possible solutions with respect to the quadrants of a plane above the magnetic source center were obtained, and the unique solution can be determined by the actual orientation and measured data. The results show that after error correction of the magnetic gradient tensor system, the positioning accuracy of a small-scale magnet (diameter of 5 cm, thickness of 0.5 cm) can be controlled within a root mean square error of 5 cm. Compared with the Euler inversion method, the proposed method has a greater detection distance with the same noise and exhibits a more reliable result.