Flight Instructor reported they were given a heading by ATC that put them in an area below Minimum Safe Altitude. Instructor contacted ATC and climbed to a safe altitude.

Narrative

The event occurred during an IFR flight plan training flight between ZZZ-ZZZ1 at approximately XA:50 local time. While speaking to ZZZ approach on XXX.X; we were instructed to fly heading 190 from ZZZ. We continued on this heading well past where we typically would be switched over; so I inquired when we would switch on XXX.X. I received somewhat of a response with an approach saying my tail number; but nothing after that. As we continued down I continually inquired and performed radio checks to no avail. Through my knowledge of the area I knew I was getting into an area of terrain and became concerned with our low altitude of 5000 ft. I was prepared to begin a climb to 6500 ft. to be at my MSA but; I utilized my knowledge of the area and figured out that I should likely call the ZZZ center on XXY.X. I contacted them while still monitoring XXX.X and they immediately responded with an altitude alert and instructed me to climb to 7000 ft. to clear the 6500 ft. MSA. I complied and as I heard on XXX.X approach trying to contact me. I informed them that I had contacted Center and they acknowledged it and told me to talk to them. While further talking to Center I was instructed to contact a phone number for a possible pilot deviation. I landed at ZZZ1 with no issues and contacted the number. While talking to the controller he instructed me that my action in contacting them was correct and the reason for the investigation was because of how low I was in the sector. The controller also said ZZZ TRACON would be looked into to ensure this would not happen again and the events would be reviewed.