The Bundesliga side went 1-0 up before the break via Thorgan Hazard as Barca missed a number of chances at Borussia-Park.
Arda Turan restored parity after the interval, though, before Gerard Pique netted the winner with 17 minutes left.
"We dominated the first half and created several chances," Luis Enrique said at a news conference.
"We didn't carve out as many opportunities after half-time, but overall it was a deserved win.
