We address the problem of extracting the position and the Euler angles of an underground object from measurements of the magnetic field above that object. The traditional Euler inversion positioning method requires us to make use of the high-order magnetic gradient tensor system and extract the second-order tensor data, but at the same time it will increase the sensitivity to errors and measurement noise, which reduces the positioning accuracy. To solve this problem, we propose a method that uses only first-order magnetic gradient tensor data and supplements the tensor spacial invariant relations to achieve a higher-precision positioning performance for magnetic object. We analyze the tensor invariants and the eigenvalues of the tensor matrix under a magnetic dipole source field, and derive two tensor spacial invariant relations: (1) the angle between magnetic moment and position vector is constant and can be represented by the tensor eigenvalues, and (2) the eigenvector of the absolute minimum eigenvalue is perpendicular to the magnetic moment and position vector, as well as the eigenvectors of remaining eigenvalues are coplanar to the magnetic moment and position vector. Hence, we compute a total four possible solutions of the position vector with respect to the four quadrants of a plane over the measuring center, and finally determine the true one based on the actual detection direction and measured magnetic field data. The results show that the proposed method has significantly higher detection accuracy and larger range under a same level noise condition than the Euler inversion positioning method. When positioning small-scale magnets (e.g. about 5 cm in diameter and 0.5 cm in thickness), we control the tri-axial coordinates accuracy with just the first-order magnetic gradient tensor system within 1 cm root mean square error.