From the time when Facebook has launched the Like button for web pages, many webmasters integrated that button in their website. This Like button makes the things easier for you if you wish to like them through your Facebook Profile.
In the scenario that the page, you would like to like via Facebook, has not any Facebook like button in it, you can still make the button available for that page by using the Facebook-Like Bookmarklet.
The installation of this bookmarklet is as simple as you like. Just drag the linked phrase Facebook-Like from below and drop at your browser toolbar (where the other bookmarks reside).
Facebook-Like — Drag and Drop to the browser toolbar.
You are done! Now whenever you want to like any web page, just click on the Facebook-Like bookmarklet from the browser toolbar and you should see the like button on that page.
Though, I was using another bookmarklet Like-o-matic to like pages through Facebook, it seems that the hosting site of that application is no longer available.
But the Facebook-Like bookmarklet works fine. Give it a try.
Related: Add Facebook Social Plugin: Like Box.