TensorFlow is a popular cloud computing framework that targets machine learning applications. It separates the specification of application logic (in a dataflow graph) from the execution of the logic. TensorFlow's native runtime executes the application with low overhead across a diverse set of hardware including CPUs, GPUs, and ASICs. Although the underlying dataflow engine supporting these features could be applied to computations beyond machine learning, certain design decisions limit this broader application, such as the inability for an application to differentiate between data items across concurrent requests. This paper introduces Pipelined TensorFlow (PTF), a system that extends TensorFlow's semantics to provide support for a broader variety of application logic. In particular, PTF supports applications that concurrently process finite batches of data on a single instantiation. PTF adds these semantics by partitioning the dataflow graph into a pipeline of smaller graphs and tagging each data item with metadata. These smaller graphs are separated by gates: new data structures in PTF that buffer data items between graphs and interpret the metadata to apply the new semantics. PTF's pipeline architecture executes on an unmodified TensorFlow runtime, maintaining compatibility with many existing TensorFlow library functions. Our evaluation shows that the pipelining mechanism of PTF can increase the throughput of a bioinformatics application by 4$\times$ while only increasing its latency by 0.13$\times$. This results in a sustained genome alignment and sorting rate of 321 megabases/second, using the compute and I/O resources of 20 computers.