The images are made with radio telescopes, which cuts through the dust quite easily. We have many other radio observations of Sagittarius A* [1], albeit at much lower resolutions. There are also numerous observations of Sagittarius A* in X-ray wavelengths, which is also fine because they are so energetic they simply punch through. All the dust and gas in the galaxy is transparent at most wavelengths except the visible one.

The Event Horizon Telescope is interesting because it is, in essence, a radio telescope that uses a "sensor" that is the size of the entire Earth. As such, it is able to make much higher resolution observations.

[1] https://en.wikipedia.org/wiki/Sagittarius_A*#/media/File:Clo...

[2] https://en.wikipedia.org/wiki/Sagittarius_A*#/media/File:X-R...