Incorrect handling of Software Application Programming Interfaces (APIs) errors results in bugs or security vulnerabilities that are hard to trigger during regular testing. Most of the existing techniques to detect such errors are based on static analysis and fail to identify certain cases where API return values are incorrectly handled. Furthermore, most of these techniques suffer from a very high false positive rate (≥50%), raising concerns regarding their practical use. We propose a dynamic analysis approach to detect API error handling bugs based on coverage-guided software fault injection. Specifically, we inject faults into APIs and observe how a program handles them. Our fault injection mechanism is generic and targeted to explore a given program's error handling behavior effectively. We avoid false positives by proactively filtering out crashes caused by infeasible faults. We implemented our technique in an automated pipeline called FuzzERR and applied it to 20 different programs spanning 444 APIs. Our evaluation shows that FuzzERR found 31 new and previously unknown bugs resulting from incorrect handling of API errors. Moreover, a comparative evaluation showed that FuzzERR significantly outperformed the state-of-the-art tools.