That is a difficult question because some cosmologists were successful long ago during ancient history. It is difficult to evaluate their contributions based upon what they had access to. Aristotle was a cosmologist but people sometimes forget him because his contributions although significant were so long ago. Today, it would have to be Stephen Hawking. He gathered and analyzed data and information in order to make accurate predictions.
His theories have been deemed brilliant to the most top level scientists. He started at a young age and has proven theories through his work. Stephen Hawking continued his career of studying and learning about space until his death recently. However, he would probably have continued his work about space for years later if he was still alive.
I think this will have to be Stephen Hawking because he had the vision, not just to construct brilliant theories and seek to prove them, but to popularise his theories. It is relatively rare for scientists in the genius category to bother about whether everyman has any conception of the studies and findings he is making. Hawking had that vision, and the fact that his well constructed and accessible books have been so widely read must be regarded as true success, not just success in terms of accolades and gongs from learned societies.
Clearly, that success must also be grounded in work that has stood the test of academic scrutiny and peer criticism. Hawking is universally respected and his theories universally accepted. All of this makes for a very real success.