C-172 Instructor reported that the student lost control of the aircraft upon landing. Instructor took control as the aircraft departed the runway and taxied onto the taxiway and to parking.

Narrative

I was on a visual approach with my student to Runway XX at ZZZ. We had a 09kt x-wind from 230. On approach I was guiding my student to correct for the x-wind. Our landing airspeed was approximately 7kts faster than normal landing speed in order to compensate for the wind and felt acceptable due to the length of the runway. Touchdown was slightly right of centerline. In a matter of a seconds the student abruptly used left rudder to center on the runway but his input was too aggressive so he over corrected to the right very quickly and we turned to the edge of the runway; then he over corrected again back to the left; putting us on the left main tire. I had taken over controls by turning the ailerons into the wind; neutralizing the elevator; and neutralizing the rudders pedals. I had stabilized the aircraft and had three options; 1.) Attempt to turn back to the runway and risk more skidding and likely flipping the plane over 2.) add full power and attempt a go-around while facing the runway edge at approximately a 60-70 angle and flying off the run way; or 3.) maintain directional control and ground roll into the grass. Option 3 had felt like the safest course of action; so I committed to the grass patch where we slowed to a stop. There were no personal injuries and no damage to the aircraft. I was able to taxi out of the grass onto the taxi way as instructed by tower and continued to parking. After shutting down the engine I debriefed my student on how and what transpired. He now understands that at faster airspeeds; his inputs need to be more gradual and not so abrupt. Our next instruction will be in the simulator remediating similar situations.