TOKYO, SEPT. 19 -- Japan's parliament today approved an income tax cut designed to stimulate the domestic economy. The idea is strongly backed by the Reagan administration, which hopes that greater spending by Japanese consumers will help ease Japan's nearly $60 billion trade imbalance with the United States.

The $10 billion tax cut given final approval by the Diet, as the Japanese parliament is known, is smaller than that originally sought by Prime Minister Yasuhiro Nakasone. But a more substantial overhaul of the tax system had to be abandoned in May because it was twinned with a new sales tax that proved politically unpopular and has been blamed for a major election defeat for the ruling Liberal Democratic Party.

The tax cut approved today was less controversial but still drew strong opposition because it was tied to, and largely offset by, a new 20 percent tax on small bank and postal savings accounts.

The income tax cuts will be achieved by reducing the number of brackets in the income tax system from 15 to 13. The cuts will provide the most help to middle-income salaried workers, according to government officials. The highest tax bracket would be dropped from about 70 percent to 60.