In the heated debate over inequality, the wealthy are usually portrayed as the cause rather than the solution. But CNBC's first-ever Millionaire Survey reveals that 51 percent of American millionaires believe inequality is a "major problem" for the U.S., and of those, nearly two-thirds support higher taxes on the wealthy and a higher minimum wage as ways to narrow the wealth gap.

The findings show that—far from being a purely self-interested voting bloc—American millionaires have complicated views when it comes to the wealth gap and opportunity in America. They are unashamed of their own wealth and attribute their success to hard work, smart investing and savings. They also believe that anyone in America can get wealthy if they work hard. Yet millionaires also believe that cultural and family issues prevent many Americans from climbing the wealth ladder. They advocate improved education, higher taxes on the wealthy and better savings incentives for the poor and middle class as important changes that would reduce inequality. Read More This hedge fund chief made $3.5 billion in 2013

The CNBC Millionaire Survey polled 514 people with investable assets of $1 million or more, which represents the top 8 percent of American households. The respondents came from around the country and were split between Democrats, Republicans and Independents. The online survey was conducted in March by Spectrem Group on behalf of CNBC. The overall margin of error was 4.3 percentage points.

When asked about the No. 1 factor in obtaining their wealth, the millionaires ranked hard work first (23 percent), followed by smart investing (21 percent) and savings (18 percent). Education ranked fourth, at 10 percent, followed by frugality and then inheritance. Only 1 percent cited luck as the top reason for their wealth. Multimillionaires, or people worth $5 million or more, were more likely to cite "running my own business" as their top wealth factor. Women were three times more likely to cite inheritance as their top wealth factor (15 percent vs. 5 percent for men), while men were more likely to cite savings (20 percent vs. 14 percent for women). Read MoreFinancial tips for 20-somethings Most millionaires still believe in the American dream. Fully 94 percent said the American dream is achievable. When asked to define that dream, the largest number (45 percent) said the American dream is "prosperity and upward mobility through hard work." Only 18 percent defined it as "spiritual and temporal happiness more than material goods." Multimillionaires, however, were far more likely to define the dream as material rather than spiritual (63 percent vs. 4 percent).

Despite being winners in the new economy, U.S. millionaires view inequality as a problem. More than half of millionaires and multimillionaires agreed that "inequality of wealth in our nation is a major problem." Yet they don't see themselves as a cause. Fully 81 percent said they don't feel embarrassed by their wealth, "because I earned it," and only 5 percent said they feel a sense of guilt about the wealth they possess. More than half said anyone in the U.S. can become wealthy if they work hard.

When asked about the reasons for inequality, most (78 percent) said the wealthy have greater access to education. Two-thirds cited that the "lack of financial literacy" prevents poor households from making better financial decisions. Read MoreSheila Bair: Money lessons for your kids More than half said cultural issues and broken families also prevent people from attaining wealth. Only 6 percent said that people worth less do not work as hard as those with wealth. The best way to reduce inequality, millionaires say, is through improved education. Fully 83 percent supported an increase in educational opportunities for the less wealthy. An equal number (64 percent) supported better savings incentives for the less wealthy and higher taxes for the wealthy. Perhaps surprisingly, a majority (63 percent) also support a minimum wage. Only 13 percent supported reducing unemployment benefits to encourage more work as a solution to inequality.