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Convolutional Neural Networks (CNNs) are the state-of-the-art solution for many deep learning applications. For maximum scalability, their computation should combine high performance and energy efficiency. In practice, the convolutions of each CNN layer are mapped to a matrix multiplication that includes all input features and kernels of each layer and is computed using a systolic array. In this work, we focus on the design of a systolic array with configurable pipeline with the goal to select an optimal pipeline configuration for each CNN layer. The proposed systolic array, called ArrayFlex, can operate in normal, or in shallow pipeline mode, thus balancing the execution time in cycles and the operating clock frequency. By selecting the appropriate pipeline configuration per CNN layer, ArrayFlex reduces the inference latency of state-of-the-art CNNs by 11%, on average, as compared to a traditional fixed-pipeline systolic array. Most importantly, this result is achieved while using 13%-23% less power, for the same applications, thus offering a combined energy-delay-product efficiency between 1.4x and 1.8x.