I managed to win using this technique.
I managed to win by using this technique.
Which is correct? And why?
Both are grammatically correct depending on your intended meaning. Your first sentence says "I won while I was using this technique" and the second says "this technique was the reason I won."
Take the following examples:
I managed to win with one hand tied behind my back.
This sentence says "I won in spite of having one hand tied behind my back".
I managed to win by tying one hand behind my back.
This one implies that tying one hand behind your back actually was the reason for winning.
So, essentially, it comes down to what you are trying to say. Using "by" emphasizes that it was the reason for winning.
Generically, "by" shows how something was done. If you omit it, you are just stating that it was done.