if you actually made a backup of the disk by using Disk Utility.
Quote from Apple:
This alternative backup process will produce a disk image of your entire Mac OS X disk's contents. This process preserves the unique attributes of your files, such as permissions, ACLs and UUIDs. A disk image backup is good for an archive-type backup for offsite storage.
UnQuote
Unless you did that procedure before replacing the drive, you will have no success in restoring the drive as there is no Image for DU to use as the Restore media.

Using Time Machine, you "should" be able to do the following:
Double click the new drive to open it.
Once its window is open, launch Time Machine from the icon in the dock. You "should" see the desktop of the failed drive. Select all and Restore.

P