Open-set test-time adaptation (OSTTA) addresses the challenge of adapting models to new environments where out-of-distribution (OOD) samples coexist with in-distribution (ID) samples affected by distribution shifts. In such settings, covariate shift-for example, changes in weather conditions such as snow-can alter ID samples, reducing model reliability. Consequently, models must not only correctly classify covariate-shifted ID (csID) samples but also effectively reject covariate-shifted OOD (csOOD) samples. Entropy minimization is a common strategy in test-time adaptation to maintain ID performance under distribution shifts, while entropy maximization is widely applied to enhance OOD detection. Several studies have sought to combine these objectives to tackle the challenges of OSTTA. However, the intrinsic conflict between entropy minimization and maximization inevitably leads to a trade-off between csID classification and csOOD detection. In this paper, we first analyze the limitations of entropy maximization in OSTTA and then introduce an angular loss to regulate feature norm magnitudes, along with a feature-norm loss to suppress csOOD logits, thereby improving OOD detection. These objectives form ROSETTA, a $\underline{r}$obust $\underline{o}$pen-$\underline{se}$t $\underline{t}$est-$\underline{t}$ime $\underline{a}$daptation. Our method achieves strong OOD detection while maintaining high ID classification performance on CIFAR-10-C, CIFAR-100-C, Tiny-ImageNet-C and ImageNet-C. Furthermore, experiments on the Cityscapes validate the method's effectiveness in real-world semantic segmentation, and results on the HAC dataset demonstrate its applicability across different open-set TTA setups.