Your solution is often used for this kind of problem. However, I would argue that it depends on the data if it truly makes sense. Let me give you two examples you can still find on kaggle.
If you wanted to mask certain parts of satellite images, you would probably get away with this approach without a drop in accuracy. These images are highly repetitive and there's likely no correlation between the segmented area and where in the original image it was taken from.
If you wanted to segment a car from its background, it wouldn't be desirable to break it into patches. Over several layers the network will learn the global distribution of a car in the frame. It's very likely that the mask is positive in the middle and negative in the corners of the image.
Since you didn't give any specifics what you're trying to solve, I can only give a general recommendation: Try to keep the input images as large as your hardware allows. In many situation I would rather downsample the original images than breaking it down into patches.
Concerning the recommendation of curio1729, I can only advise against training on small patches and testing on the original images. While it's technically possible thanks to fully convolutional networks, you're changing the data to an extend, that might very likely hurt performance. CNNs are known for their extraction of local features, but there's a large amount of global information that is learned over the abstraction of multiple layers.