Large retailers are complying with the Labor Department’s new overtime requirements more than a month before enforcement begins, even as their lobbyists try to block the rule in Congress.

The regulation, expected to make some 4 million new workers eligible for overtime pay starting Dec. 1, is a signature piece of the president’s middle-class agenda. The rule is aimed particularly at the retail sector, where managers and assistant managers are known to work long hours in the $30,000-$50,000 salary range, without the time-and-a-half overtime premium afforded to their hourly co-workers.

The biggest retail corporation of them all, Wal-Mart Stores Inc.,...