I have always been using this method:

from sys import argv


and use argv with just argv. But there is a convention of using this:

import sys


and using the argv by sys.argv

The second method makes the code self documented and I (really) adhere to it. But the reason I prefer first method is it is fast because we are importing only the function that is needed rather than import the whole module (which contains more useless functions which python will waste time importing them). Note that I need just argv and all other functions from sys are useless to me.

So my questions are. Does the first method really makes the script fast? Which method is preferred most? Why?

Importing the module doesn't waste anything; the module is always fully imported (into the sys.modules mapping), so whether you use import sys or from sys import argv makes no odds.
The only difference between the two statements is what name is bound; import sys binds the name sys to the module (so sys -> sys.modules['sys']), while from sys import argv binds a different name, argv, pointing straight at the attribute contained inside of the module (so argv -> sys.modules['sys'].argv). The rest of the sys module is still there, whether you use anything else from the module or not.
There is also no performance difference between the two approaches. Yes, sys.argv has to look up two things; it has to look up sys in your global namespace (finds the module), then look up the attribute argv. And yes, by using from sys import argv you can skip the attribute lookup, since you already have a direct reference to the attribute. But the import statement still has to do that work, it looks up the same attribute when importing, and you'll only ever need to use argv once. If you had to use argv thousands of times in a loop, it could perhaps make a difference, but in this specific case it really does not.
Hence, the choice between one or the other should be based solely on coding style.
In a large module, I'd certainly use import sys; code documentation matters, and using sys.argv somewhere in a large module makes it much clearer what you are referring to than just argv ever would.
If the only place you use argv is in a '__main__' block to call a main() function, by all means use from sys import argv if you feel happier about that:
if __name__ == '__main__':
    from sys import argv
    main(argv)

I'd still use import sys there myself. All things being equal (and they are, exactly, in terms of performance and number of characters used to write it), that is just easier on the eye for me.
If you are importing something else altogether, then perhaps performance comes into play. But only if you use a specific name in a module many times over, in a critical loop for example. But then creating a local name (within a function) is going to be faster still:
 import somemodule

 def somefunction():
      localname = somemodule.somefunctionorother
      while test:
          # huge, critical loop
          foo = localname(bar)

