Filed on Monday, the federal lawsuit names the five boy band members together with Simon Cowell's management company Syco and record label Sony Music Entertainment.
Attorneys for the California band are seeking an injunction that would stop Syco Entertainment and Sony Music from using the name One Direction in promotional materials. They also want a share of the profits earned by the chart-topping British boys.
According to the lawsuit, the rock band has been using the name since 2009, recorded two albums and filed an application to register the trademark name in the US in February 2011.
They say they are entitled to three times the profits made by One Direction, as well as compensatory damages in excess of $1m.
The lawsuit claims that Syco and Sony Music "chose to ignore the plaintiff's rights and wilfully infringed them" after they realised in early 2011 that the two bands share the same name.
Syco declined to comment on Tuesday, and a representative for Sony Music could not immediately be reached.
The five-man California band is led by singer and pianist Sean O'Leary. According to their website, its members began making music at school and the group has played at local fairs and bars.
The lawsuit said the continued use by both bands of the same name was causing "substantial confusion and substantial damage" to the goodwill earned by the California group.