An executable utility is injected into cloud-based virtual machines (VMs) that are subject to backups by a data storage management system tasked with protecting the cloud-based VMs and their associated data. The utility is injected into a target VM which is “live” and operating. The utility analyzes the VM's live volume to discover data extents therein, and for each extent computes a respective checksum and determines whether the extent is a “hole.” Afterwards, checksums help identify changed data in successive snapshots of the live volume, so that only changed data will be read and backed up in incremental backups. Time is saved in performing the backup operation first by pre-warming the backup's source volume in parallel with the utility analyzing the live volume, and second by skipping read operations for extents unchanged since a preceding backup. The resulting incremental backup operation is sped up as compared to prior art approaches.