We compute a basis for the $p$-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give $p$-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.