The mechanics of DNA supercoiling is a subject of crucial importance to uncover the mechanism and kinetics of several enzymes. It is therefore being investigated using several biochemical and biophysical methods including single molecule experimental techniques. An interesting problem within this realm is that of torsional buckling and plectoneme formation in DNA as it is simultaneously put under tensile and torsional stress. Analytical solutions to this problem are difficult to find since it involves nonlinear kinematics and thermal fluctuations. In this paper we use ideas from the Kirchhoff theory of filaments to find semi-analytical solutions for the average shape of the fluctuating DNA under the assumption that there is no self-contact. The basic step in our method consists of combining a helical solution of the rod with a non-planar localizing solution in such a way that the force, moment, position and slope remain continuous everywhere along the rod. Our solutions allow us to predict the extension vs. linking number behavior of long pieces of DNA for various values of the tension and temperature. An interesting outcome of our calculations is the prediction of a sudden change in extension at buckling which does not seem to have been emphasized in earlier theoretical models or experiments. Our predictions are amenable to falsification by recently developed single molecule techniques which can simultaneously track the force–extension as well as the torque–rotation behavior of DNA.