An apparatus comprises at least a first container host device implementing a plurality of containers, a storage platform coupled to the first container host device and implementing storage resources for utilization by the containers, and a container storage controller associated with the first container host device. The container storage controller is configured to provision portions of the storage resources for respective ones of the containers including for each of the containers at least one storage volume. The provisioned storage volume for a given one of the containers is partitioned into at least a data storage volume and a state storage volume, with the data storage volume being configured to store data for at least one application running in the given container, and the state storage volume being configured to store state information of the given container for use in migrating the given container from the first container host device to a second container host device.