Huge success. I didn't want to pay for a software like Acronis True Image, so I used the freeware Clonezilla.
Used a USB flash drive to boot into Clonezilla, and used an external HDD to store the clone image. Copied the image back to the new, larger HDD.
Booted into new HDD. Windows loaded as normal, and everything looked the same way... including the partitions! New drive is about 200GB bigger, so there was 200GB of unused, unpartitioned space on new HDD.
Couldn't increase the partition und