In a cross-section of 40 countries in 1995, religion (measured as the percentage of people attending religious service at least once a week, from World Value Survey) is positively correlated with corruption (measured from the index developed by Transparency International). There seem to be three clusters of countries within the scatterplot reported below: countries with low levels of religiosity and low levels of corruption (e.g., New Zealand); countries with intermediate levels of religiosity and medium-to-high levels of corruption (e.g., Brazil); godless and corrupt countries (e.g., China). One should remark that all very religious countries are also very corrupt. However, high levels of corruption are associated both with low and high religiosity, indicating that the impact of religion on corruption is fragile. One may think that in some of the countries in the NE region of the graph attendance of religious services is low because it was/is actively discouraged by the government (e.g., China, Russia). The results are however not different when I use a definition of religiosity based on personal beliefs rather than active behavior. In any case, the link disappears after controlling for GDP per capita.
In conclusion, I fail to add a statistical significant impact on corruption to the list of bads brought by religion, but it is quite safe to claim that the level of religiosity in a country does not incoculate it against high levels of corruption. Italy being a point in case.