Last week, Burnley failed to break down a dogged London team's defence, having drawn 0-0 with Queens Park Rangers at Turf Moor.
This was a similar story. And worryingly for Scott Parker, his side have now failed to score in three of their past five league matches.
Burnley started the game brightly, with Luca Koleosho and Andreas Hountondji looking particularly threatening but without troubling Millwall keeper Lukas Jensen.
A long stoppage for an injury to assistant referee Matthew Smith seemed to stunt their momentum and it was the hosts who finished the first half well, with Langstaff forcing a stunning diving save from James Trafford and the brilliant Azeez volleying over the bar.
Burnley pushed for the equaliser with long periods of possession in the Millwall half but without managing to create chances of note.
Substitute Jay Rodriguez headed over late on for the visitors but Burnley were unable to come up with the goal that would continue an unbeaten run that had stretched back to their Carabao Cup defeat by Wolves on Wednesday, 28 August.