Cessna 172 Instructor pilot reported an engine malfunction during training maneuvers with a student pilot. The Instructor returned to the departure airport and landed safely.

Narrative

Our private pilot training flight began as normal. Preflight was performed; oil quantity was verified; and so was fuel quantity/quality. We began taxiing and performed our run-up check as usual; nothing looked out of order and was within operating specifications. We continued our flight and taxied to the hold short line to depart. During the beginning of the flight lesson; the aircraft performed correctly. Towards the end of the lesson; the student was setting up for a POWER OFF STALL when we started experiencing engine roughness and vibrations about 14nm from ZZZZ. At this point; I took control of the aircraft and troubleshooted. I decided to go back to ZZZZ; since the oil pressure and temperature were inside operating specifications and the aircraft was still producing thrust to maintain our airspeed and altitude. During our way back; the engine roughness continued. Every time power was increased; the engine roughness increased; and so did the vibrations. So; I opted to reduce our power setting to one that would allow us to maintain our altitude and airspeed but reduce the vibrations. During our approach into ZZZZ; 10 nautical miles out to the west of the airport; I contacted the Tower Controller at ZZZZ and notified him that we were coming in for a full stop and that we were experiencing engine roughness and vibrations. At that point; the Tower Controller cleared us to land on Runway X. Abeam the water tank; I decided to hold our altitude at 1500 feet in case we lost our engine; there was enough gliding range to reach a safe landing area. By doing this; I made the mistake of entering ZZZZ1 airspace; which has a floor at 1199 feet; without having two-way communications. As soon as I noticed I had busted Class C; I descended to 1110 feet. The bottom line is that I made a mistake; the correct action should have been to contact ZZZ Approach so they would have cleared me through Class C for me to maintain a safe altitude and separation from any other traffic. This is my first time in this type of situation. Next time; in any type of situation; I'll make sure to pay more attention to what type of airspace I'm going to be flying in and comply with the requirements.