Test-time adaptation (TTA) aims at adapting a model pre-trained on the labeled source domain to the unlabeled target domain. Existing methods usually focus on improving TTA performance under covariate shifts, while neglecting semantic shifts. In this paper, we delve into a realistic open-set TTA setting where the target domain may contain samples from unknown classes. Many state-of-the-art closed-set TTA methods perform poorly when applied to open-set scenarios, which can be attributed to the inaccurate estimation of data distribution and model confidence. To address these issues, we propose a simple but effective framework called unified entropy optimization (UniEnt), which is capable of simultaneously adapting to covariate-shifted in-distribution (csID) data and detecting covariate-shifted out-of-distribution (csOOD) data. Specifically, UniEnt first mines pseudo-csID and pseudo-csOOD samples from test data, followed by entropy minimization on the pseudo-csID data and entropy maximization on the pseudo-csOOD data. Furthermore, we introduce UniEnt+ to alleviate the noise caused by hard data partition leveraging sample-level confidence. Extensive experiments on CIFAR benchmarks and Tiny-ImageNet-C show the superiority of our framework. The code is available at https://github.com/gaozhengqing/UniEnt.