Jay Rodriguez's eighth goal of the season earned Burnley victory at Millwall and secured back-to-back league wins for Eddie Howe's side.
Neither side threatened much until Rodriguez steered in a well-worked corner on 37 minutes.
Millwall left the pitch to loud boos at the interval but improved dramatically.
John Marquis had a strike ruled out for offside and had an effort cleared off the line while James Henry also went close but the visitors held on.
The result means the Lions' winless run extends to eight games and they have also failed to score in three consecutive home matches.
But the pressure has eased on Clarets manager Howe whose also masterminded a thumping 5-1 win over Nottingham Forest in midweek.
Howe's men frustrated the hosts in an uninspiring first half - notable only for Rodriguez's opener when he met Ross Wallace's corner and made enough contact to send the ball past David Forde.
Dany N'Guessan fired Millwall's best chance of a disappointing half wide but they showed far more purpose in the early stages of the second half, with Henry's threatening shot blocked and substitute Marquis being denied a leveller by an assistant referee's flag when he slid the ball into the net.
Brian Howard's curling free-kick then saw Lee Grant produce a fine a sprawling save and the keeper also denied Liam Feeney.
Millwall continued to threaten and Hamer Bouazza climbed off the bench to rifle a late effort narrowly over and Marquis saw a shot cleared off the line, but although Burnley rarely looked like adding to their lead, they held on for the win.