I asked a question and got this reply which helped.
UPDATE TABLE_A a JOIN TABLE_B b ON a.join_col = b.join_col AND a.column_a = b.column_b SET a.c
Alternative way of achieving same result is not to use JOIN keyword at all.
JOIN
UPDATE TABLE_A, TABLE_B SET TABLE_A.column_c = TABLE_B.column_c + 1 WHERE TABLE_A.join_col = TABLE_B.join_col