Aim To identify whether eradication or containment is expected to be the most cost-effective management goal for an isolated invasive population when knowledge about the current extent is uncertain. Location Global and South Africa. Methods We developed a decision analysis framework to analyse the best management goal for an invasive species population (eradication, containment or take no action) when knowledge about the current extent is uncertain. We used value of information analysis to identify when investment in learning about the extent will improve this decision-making and tested the sensitivity of the conclusions to different parameters (e.g. spread rate, maximum extent, and management efficacy and cost). The model was applied to Acacia paradoxa DC, an Australian shrub with an estimated invasive extent of 310ha on Table Mountain, South Africa. Results Under the parameters used, attempting eradication is cost-effective for infestations of up to 777ha. However, if the invasion extent is poorly known, then attempting eradication is only cost-effective for infestations estimated as 296ha or smaller. The value of learning is greatest (maximum of 8% saving) when infestation extent is poorly known and if it is close to the maximum extent for which attempting eradication is optimal. The optimal management action is most sensitive to the probability that the action succeeds (which depends on the extent), with the discount rate and cost of management also important, but spread rate less so. Over a 20-year time-horizon, attempting to eradicate A. paradoxa from South Africa is predicted to cost on average ZAR 8 million if the extent is known, and if our current estimate is poor, ZAR 33.6 million as opposed to ZAR 32.8 million for attempting containment. Main conclusions Our framework evaluates the cost-effectiveness of attempting eradication or containment of an invasive population that takes uncertainty in population extent into account. We show that incorporating uncertainty in the analysis avoids overly optimistic beliefs about the effectiveness of management enabling better management decisions. For A. paradoxa in South Africa, attempting to eradicate is likely to be cost-effective, particularly if resources are allocated to better understand and improve management efficacy.