Robustness evaluation against adversarial examples has become increasingly important to unveil the trustworthiness of the prevailing deep models in natural language processing (NLP). However, in contrast to the computer vision domain where the first-order projected gradient descent (PGD) is used as the benchmark approach to generate adversarial examples for robustness evaluation, there lacks a principled first-order gradient-based robustness evaluation framework in NLP. The emerging optimization challenges lie in 1) the discrete nature of textual inputs together with the strong coupling between the perturbation location and the actual content, and 2) the additional constraint that the perturbed text should be fluent and achieve a low perplexity under a language model. These challenges make the development of PGD-like NLP attacks difficult. To bridge the gap, we propose TextGrad, a new attack generator using gradient-driven optimization, supporting high-accuracy and high-quality assessment of adversarial robustness in NLP. Specifically, we address the aforementioned challenges in a unified optimization framework. And we develop an effective convex relaxation method to co-optimize the continuously-relaxed site selection and perturbation variables and leverage an effective sampling method to establish an accurate mapping from the continuous optimization variables to the discrete textual perturbations. Moreover, as a first-order attack generation method, TextGrad can be baked into adversarial training to further improve the robustness of NLP models. Extensive experiments are provided to demonstrate the effectiveness of TextGrad not only in attack generation for robustness evaluation but also in adversarial defense. From the attack perspective, we show that TextGrad achieves remarkable improvements in both the attack success rate and the perplexity score over five state-of-the-art baselines. From the defense perspective, TextGrad-enabled adversarial training yields the most robust NLP model against a wide spectrum of NLP attacks.