This is my view.
Asians experienced a lot of racism when they first came to this part of the world -- probably more than you think.
But they were immigrants, and immigrants are self-selected for traits that tend to facilitate success. True, laws were eventually passed to block any more from coming, but their immigrant status would nonetheless provide the foundation for future generations.
Importantly, they never faced quite the same degree of racism that blacks faced, and they were also a smaller group (i.e. they posed less of a threat), and over time this allowed them to build on their initial advantage as immigrants. What I mean is that as they were treated better than blacks, and allowed greater social intercourse with whites, this allowed them to interact more with whites, and in this way whites came to regard them as 'not-so bad,' which in turn gave them more opportunities, the taking advantage of which further enhanced their perception in the eyes of whites. It is a virtuous circle. Later, after immigration laws were relaxed, more educated and or -education-minded Asians arrived, and these newer groups did even more to cement the popular perception of them.
Key to the process is that whites abandoned their stereotypes of Asians as they had more contact with them.
Arguably, a similar thing may be possible for blacks, but I would contend that there are far more obstacles, the biggest being that we are more phenotypically different from whites than Asians are from them. Roughly speaking, this means that we can expect some form of unequal treatment for the foreseeable future, and possibly forever. Ultimately, I don't believe that blacks will ever be able to go the Asian route of white acceptance. Improvements can be made, however. Anti-black stereotypes, as deeply entrenched as they are, are not completely inflexible over the long-term. The more enduring -- and so more serious -- problem is one of value.