Monitoring and detecting abnormal events in cyber-physical systems is crucial to industrial production. With the prevalent deployment of the Industrial Internet of Things (IIoT), an enormous amount of time series data is collected to facilitate machine learning models for anomaly detection, and it is of the utmost importance to directly deploy the trained models on the IIoT devices. However, it is most challenging to deploy complex deep learning models such as Convolutional Neural Networks (CNNs) on these memory-constrained IIoT devices embedded with microcontrollers (MCUs). To alleviate the memory constraints of MCUs, we propose a novel framework named Tiny Anomaly Detection (TinyAD) to efficiently facilitate onboard inference of CNNs for real-time anomaly detection. First, we conduct a comprehensive analysis of depthwise separable CNNs and regular CNNs for anomaly detection and find that the depthwise separable convolution operation can reduce the model size by 50-90% compared with the traditional CNNs. Then, to reduce the peak memory consumption of CNNs, we explore two complementary strategies, in-place, and patch-by-patch memory rescheduling, and integrate them into a unified framework. The in-place method decreases the peak memory of the depthwise convolution by sparing a temporary buffer to transfer the activation results, while the patch-by-patch method further reduces the peak memory of layer-wise execution by slicing the input data into corresponding receptive fields and executing in order. Furthermore, by adjusting the dimension of convolution filters, these strategies apply to both univariate time series and multidomain time series features. Extensive experiments on real-world industrial datasets show that our framework can reduce peak memory consumption by 2-5x with negligible computation overhead.