PA-28 Flight Instructor reported engine roughness on climb-out; took controls from the Student Pilot; and performed an air turnback.

Narrative

Intro flight with new Student Pilot. The student was climbing at 80 kt. when there was a loud crack and the engine began to run roughly. We leveled the aircraft; took controls from student; all engine instruments green; mixture full rich; fuel in both tanks; on proper tank; carb heat on; [and] began to turn towards ZZZ. [Requested priority handling]; requested landing. Cleared to land; changed to [Runway] XXL from [Runway] XY. Tower asked what happened. Rough engine; unable to resolve. Tower asked change to [Runway] XXR. [Runway] XXR cleared to land. Kept power; despite rough engine. Pulled power to idle on final; still rough. Tower allowed taxi to terminal. Emergency services declined. Standard engine shutdown.Upon inspection; exhaust pipe had snapped in half; alerted Maintenance. In the interest of aviation safety; a landing was conducted with no damage to aircraft; persons; or property. As an instructor; I can attest to the value of proper practices and procedures. Tower called Chief Instructor and praised response and actions.