How to restore Mac OS X on a new hard-drive?
Yes a Time Machine (TM) backup is sufficient; you don't need to "install" the OS and then restore the backup.  However you will need a copy of the OS at least and that can definitely be placed on either a USB stick or external HDD. The OS media contains the utilities you'll need to restore the TM backup.  Example here.

(edit for clarity) Depending on how new your MBP is, more specifically which version of OS X you have, you may also have a built in Recovery that has all the tools you need to do this as well.