The District moved this week to update its minimum wage for contractors — a long-overdue move that could entitle hundreds of workers to significant back pay.

A 2006 law requires companies working under city contracts of $100,000 or more to pay their workers a “living wage” that is set every year based on the city’s cost of living. But the District’s employment services agency did not reset the rate between 2010 and Monday, when it announced a 90-cent increase — from $12.50 an hour to $13.40 an hour.

The department also announced that, for the first time, the rate will apply to many home health workers. Many of those workers care for the city’s mentally ill or developmentally disabled residents under government contracts funded through Medicaid.

A policy dispute over whether those workers were subject to the wage requirement contributed to the delay in the update of the wage rate.

At an October hearing, the official in charge of city’s system of care for disabled residents said mandating higher wages could cause some providers to shut down and “create a real crisis in the service system.”

Those concerns appear to have been addressed. Wayne Turnage, the city’s director of health care finance, said Monday that he expects federal Medicaid authorities to approve the higher contract costs expected because of the wage update.

But, he said, providers will not get retroactive payments to cover higher costs, even though the new wage increase is retroactive to January 2013. Under Monday’s announcement, an employee working a standard week at the previous minimum wage is entitled to more than $1,800 in back pay.

Employment officials said employers can request a waiver from the back-pay requirement if they can prove “significant financial hardship.”

Advocates praised the move and called on the city to make sure employees receive money they are owed. Ari Weisbard, deputy director of the Employment Justice Center, said his group was “heartened” to see Mayor Vincent C. Gray (D) take action after months of prodding.

“We hope the mayor will be more proactive in budgeting for and implementing other worker protection laws, like the minimum wage and paid sick leave law that just passed,” he said, referring to the D.C. Council’s recent move to mandate an $11.50 minimum wage for all District employers by 2016. City government contractors will remain bound by the higher standard.

The new contractor rate was issued a day before the council was set to vote on a resolution calling on the Gray administration to take action on the matter.

Council member Vincent B. Orange (D-At Large), who presided over the October hearing, modified the resolution Tuesday to call for another rate reset in March, when another update is due by law. “We are watching this, and we are expecting the calculation to be made,” he said.

The District’s finance office estimates that the higher wage could cost the city at least $13 million a year in additional contract costs.

Orange said the increased cost was justified, especially considering the city’s burgeoning coffers — reflected recently in a $20 million upgrade in projected city revenue.

“Those are artificial surpluses,” he said. “That money belongs to the people who have done the work.”