We introduce a function on sequences, which we call the Pfaffian transform, using the Pfaffian of a skew-symmetric matrix. We establish several basic properties of the Pfaffian transform, and we use the transfer matrix method to show that the set of sequences with rational generating functions is closed under the Pfaffian transform. We conclude by computing the Pfaffian transform of a variety of sequences, including geometric sequences, the sequence of Fibonacci numbers, the sequence of Pell numbers, the sequence of Jacobsthal numbers, and the sequence of Tribonacci numbers. Throughout we describe a generalization of our results to Pfaffians of skew-symmetric matrices whose entries satisfy a Pascal-like relation.