What's the easiest way to see this? The only thing I could think to do was try to patch together trivializations. I couldn't find a way to make that work. Thank you!
edit: For the record, here's why I asked about this special case of the more general result about fiber bundles over contractible spaces. In the much beloved book by Bott and Tu, it's claimed that the Leray Hirsch theorem can be proved in the same way the Kunneth theorem is proved: Induct on the size of a finite good cover for the base space, applying the Mayer Vietoris sequence and the Poincare lemma for the induction step. Its assumed that there exists a finite good cover for the base space but it's not assumed this cover is a refinement of the cover of the base space which gives the local trivializations of the fiber bundle. Therefore to apply the Poincare lemma in the induction step it seems that you need to know that the result I asked about is true. Since fiber bundles had just been introduced in the text I thought may be there was a short, elementary proof that the authors had taken for granted.