As per my knowledge, if we can find the Defects-Per-Million-Opportunities from a Normal Distribution, then we can find the sigma level.
But if we have exponential distribution, how can we find the sigma level?
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This is a case where less known fact is applicable - six sigma does not have to be applicable to data with normal distribution. Or at least not raw data.
The answer you are looking for is in Central Limit Theorem.
The Central Limit Theorem is one of the cornerstones of lean six sigma and one of the first concepts understood in lean six sigma statistical training. We know that as we collect data from a process, the data points may or may not represent a normal distribution. However, when large enough (n>30) samples are drawn from any distribution, the distribution of the sample means approaches a normal one.
So in other words, you draw samples from your dataset and regardless of dataset distribution mean values of these samples will have normal distribution.