The accurate segmentation of distinct cardiac regions from cardiac magnetic resonance (CMR) images is pivotal for enhancing the diagnosis and prognosis of heart diseases. However, the complex structure of the heart and the low contrast in CMR images pose significant challenges to achieving precise segmentation. In response to these difficulties, we propose a novel approach named BSANet, which integrates the inherent local characteristics of convolution with the global feature extraction of the Transformer through an elaborately designed multi-scale Boundary-Aware (MBA) module and a Scale-Aggregation TransFormer (SAT) module. The MBA module is specifically tailored to address the issue of blurry boundaries in CMR images. It enhances the edge extraction ability within low-level feature maps. On the other hand, the SAT module is designed to tackle the complexity and diversity of heart structures. This module retains crucial scale information to enhance segmentation performance while efficiently leveraging the computational capabilities of the Transformer via spatial reduction. Experimental results performed on three datasets show the superior performance of BSANet compared to the state-of-the-art methods. Notably, on the ACDC dataset, the proposed BSANet achieves the highest Dice score of 92.39%, while maintaining the lowest parameters. Furthermore, BSANet demonstrates optimal generalization performance on the RVSC and Sunnybrook datasets, underscoring its potential as a valuable tool for radiologists in clinical diagnosis.