Jay Rodriguez claimed his eighth goal of the season as Burnley picked up all three points away at Millwall.
Neither side had a shot on target until Rodriquez scored from a corner after 39 minutes.
Hosts Millwall controlled large parts of the second half, but visiting goalkeeper Lee Grant produced saves from Brian Howard and Liam Feeney to ensure Eddie Howe's side returned north with the victory.
Kenny Jackett's Millwall now have a winless run which stretches back eight matches.