The 44-year-old replaces Chris Powell, who was sacked on Wednesday with Town 18th in the Championship table.
He had previously been linked with rejoining ex-Dortmund boss Jurgen Klopp on the coaching staff at Liverpool.
Academy manager Mark Lillis will be in charge of the team for Saturday's Championship derby against Leeds, with Wagner to take over on Monday.
"David's football philosophy is directly in line with ours; he fits for what we need," Huddersfield chairman Dean Hoyle told the club website.
"It is a new departure for us. He will be head coach; he is the club's first from outside the UK and Ireland and he brings a new approach and new ideas.
"In his own words, and as we hope will be seen in practice, his playing style echoes the one that has brought Borussia Dortmund such success over the last five years."
Wagner quit his role as Dortmund's under-23 manager at the start of this month after four years.
The former United States international helped Borussia Dortmund II, Dortmund's reserve side, to promotion to the German third tier in 2012.
They were relegated back to the regionalised fourth tier last season.
Huddersfield Town have appointed former Borussia Dortmund coach David Wagner as their new head coach.