The Republican governor of Ohio, the Democratic mayor of Cleveland and the local teachers union have united to overhaul how teachers are hired, fired and paid, a rare example of cooperation in education that some critics warn could still face challenges in the implementation.

The overhaul, signed into law by Gov. John Kasich this month, will allow the district to link teachers' pay, in part, to student test scores, and to lay off teachers based on performance instead of seniority....