Extreme weather events have significant adverse costs for individuals, firms, communities, regional, and national economies. Extreme Event Attribution (EEA), a methodology that examines the degree to which anthropogenic greenhouse gas emissions had changed the occurrence of specific extreme weather events, allows us to quantify the climate-change-induced component of these costs. We use EEA to aggregate the global economic damage from extreme weather events that is attributable to anthropogenic climate change. For that, we collect data from all available attribution studies which estimate the Fraction of Attributable Risk (FAR) for extreme events, and combine these FAR estimates with data on the socio-economic costs of these events. With extrapolation for missing data, we then arrive at our benchmark estimates. We find that US$ 143 billion per year, of the costs of extreme events during the last twenty years, is attributable to anthropogenic climatic change. This EEA-based method for calculating the costs of climate change from extreme weather differs fundamentally from other approaches to climate cost estimation. Those other approaches use macroeconomic modelling embedded within climate models in various types of Integrated Assessment Models (IAM). As we show, our research is not directly comparable, but it does provide a new form of evidence that suggests that most IAMs are substantially under-estimating the current economic costs of climate change. Given some of the data deficiencies we identify in terms of temporal and spatial coverage, the purpose here is not to produce a definitive quantification, but rather to sketch a path towards a more comprehensive and reliable estimation. As better EEA studies and more thorough and exhaustive economic costs estimates for extreme events become available over time, and the method is refined, the precision of this approach's estimates will increase in tandem.