The transition amplitudes for passage through a potential curve crossing are calculated by means of the first order Magnus approximation to the coupled equations in the momentum representation, together with a semiclassical mapping prescription. The cases of potentials with similar or different slopes at the crossing point require a different treatment: for similar slopes the semiclassical Magnus approximation is practically identical with the exponential uniform distorted wave approximation (EUDWBA); for slopes with different signs a new probability conserving approximation is obtained, which differs from the EUDWBA and leads to a substantial improvement of the transition probabilities in the weak and intermediate coupling region. The present approximation can easily be extended to multistate crossings.