There is no good answer to this question as posed, because many Asian-American ethnicities are poorer than the U.S. general public. In fact, Asian Americans' high incomes are largely due to Indian Americans, which is not necessarily the group most Americans think of when they hear the term "Asian American."
Let's look at a 2012 Pew study of "The Rise of Asian Americans:"
The Pew Research Center survey was designed to contain a nationally
representative sample of each of the six largest Asian-American groups
by country of origin—Chinese Americans, Filipino Americans, Indian
Americans, Vietnamese Americans, Korean Americans and Japanese
Americans. Together these groups comprise at least 83% of the total
Asian population in the U.S.
The basic demographics of these groups are different on many measures.
For example, Indian Americans lead all other groups by a significant
margin in their levels of income and education. Seven-in-ten
Indian-American adults ages 25 and older have a college degree,
compared with about half of Americans of Korean, Chinese, Filipino and
Japanese ancestry, and about a quarter of Vietnamese Americans.
On the other side of the socio-economic ledger, Americans with Korean,
Vietnamese, Chinese and “other U.S. Asian” origins have higher shares
in poverty than does the U.S. general public, while those with Indian,
Japanese and Filipino origins have lower shares.
So three "Asian American" groups are more likely to be impoverished than the average American and three "Asian American" groups are not. Vietnamese Americans and "other Asian American" groups earn around $5,000/year less than the average American ($40,000/year); the average Filipino American earns about as much as the average American; the average Korean American earns $5,000 more, the average Chinese American earns $10,000 more, while Japanese Americans on average earn $15,000 more and Indian Americans about $25,000 more. Furthermore, these groups all differ from one another on plenty of other important cultural dimensions. So there is no reasonable answer to this question as posed. One would need to ask about the histories of these groups separately.
Source: Full Pew Report, 2013