COVID-19 pandemic has a devastating impact on human health and well-being. Numerous biological tools have been utilised for COVID detection, but most of the tools are costly, time-extensive and need personnel with domain expertise. Thus, a cost-effective classifier can solve the problem where cough audio signals showed potentiality as an screening classifier for COVID-19 diagnosis. Recent ML approaches on coughbased covid-19 detection need costly deep learning algorithms or sophisticated methods to extract informative features. In this paper, we propose a low-cost and efficient envelope approach, called CovidEnvelope, which can classify COVID-19 positive and negative cases from raw data by avoiding above disadvantages. This automated approach can select correct audio signals (cough) from background noises, generate envelope around the informative audio signal, and finally provide outcomes by computing area enclosed by the envelope. It has been seen that reliable data-sets are also important for achieving high performance. Our approach proves that human verbal confirmation is not a reliable source of information. Finally, the approach reaches highest sensitivity, specificity, accuracy, and AUC of 0.96, 0.92, 0.94, and 0.94 respectively to detect Covid-19 coughs. Our approach outperformed other existing models on data pre-processing and inference times, and achieved accuracy and specificity of 0.91 and 0.99 respectively, to distinguish COVID-19 coughs from other coughs, resulted from respiratory diseases. The automatic approach only takes 1.8 to 3.9 minutes to compute these performances. Overall, our approach is fast and sensitive to diagnose the people living with COVID-19, regardless of having COVID-19 related symptoms or not. In this connection, the model can be implemented easily in mobile-devices or web-based applications, and countries with poor health facilities will be highly beneficiary for covid diagnosis and measuring prognostication.