You can move the partition with a number of tools from dd to gparted. I would move it from a live CD.
Once you move the partition you need to do two things
update-grub
First mount your new root partition.
sudo mount /dev/sda1 /mnt
Prepare the chroot
for i in /dev /dev/pts /proc /sys; do sudo mount -B $i /mnt$i; done
I personally would update and re-install grub
sudo chroot /mnt
update-grub
grub-install /dev/sda
update fstab
While still in the chroot, edit /etc/fstab , make sure your root partition points to /dev/sda1
You can use "/dev/sda1" or UUID, to list your partitions by UUID use blkid
# If needed
blkid
nano /etc/fstab
Make sure your root entry looks something like
# / moved to /dev/sda1
UUID=xxxx-yyy-zzz / ext4 errors=remount-ro 0 1
Or
# / moved to /dev/sda1
/dev/sda1 / ext4 errors=remount-ro 0 1
Reboot
Reboot your computer and all should be well.
All the assumes you do not have a separate /boot partition. If you do, when configuring the chroot, mount your boot partition at /mnt/boot