There is a Chinese proverb that says that it doesn't really matter whether you think you can do something or you can't. Either way, you are right. Same applies here. If a group of people really believe they can do something better than another group, more than likely, they will.
It seems to me that you are looking for a magic recipe to "surpass established companies", whatever that means (making more money? achieving a larger market share? maybe both?). In any case, I am afraid that there isn't such a thing.
However, let me tell you what I think are the advantages of startups.
First and foremost, the people. If you are "crazy" enough to risk your capital, emotional well being, and your reputation, not to mention countless hours of hard work, to pursue something, then you have a set of powerful incentives and a tremendous advantage over an organization filled with people who... well... they are basically just waiting for another paycheck. Passion and commitment is the key.
Second, startups can adjust, tweak and refine their products and services way faster than big organizations. Think of it as driving a small car with 2 or three people vs driving a huge bus with 100 people. If you notice something is wrong, it will take little to realize what's wrong with the car. Maybe you took a wrong turn? Maybe it is one of the passengers? Maybe it is the car? On the other hand, the bus will take longer to realize that. Too many people involved driving the goddamn thing. The mechanics are more complex, and there are always a sense of invulnerability inside the bus that prevents you from seeing the naked truth.
Third, for startups is easier to offer better customer service than large organizations. The personal touch, that authenticity feeling that we all love to experience is easier to see on a startup. Capitalize on it. It works wonders.
There are also other things, some of which Nir and Lloyd S have already mentioned, so I won't repeat them here.
Finally, remember that success is a very poor teacher, and that working in a startup will expose you to invaluable experiences that will accompany you all throughout your professional life. Even if you don't achieve your goals, and even if you lose money, you will have plenty of time to recover (unless you are in your 50s or 60s). It is always better to try and fail, than to fail to try.