Corruption is eliminated by properly aligning incentives. Capitalism is also all about properly aligning incentives. Moving to a more capitalism-heavy system usually causes countries to get much richer.
Eastern Europe went through a similar transition. Before the iron curtain fell, the eastern bloc operated on favors more than it operated on money. This definitely isn't the case any more.
This is probably the best explanation. I didn't consider that when incentives are better aligned through capitalism, that perceived corruption may naturally fall. Your example of Eastern Europe is a very good one.
Eastern Europe went through a similar transition. Before the iron curtain fell, the eastern bloc operated on favors more than it operated on money. This definitely isn't the case any more.