The Clarets have now conceded just nine goals in 30 Championship matches this season, yet Matt Ritchie twice went close to breaking through their impressive defence.
At the other end, the impressive Andre Dozzell cleared of the line late on from Lucas Pires, but otherwise, the hosts admirably stood firm.
And here are our player ratings...
9. Matt Ritchie - 7
(Replaced by Cohen Bramall on 87 mins) Had the first half’s best chance after being put through the right channel by Lang, but shot wide. Then, on 85 minutes, brilliantly denied by Trafford for a potential winner. Regardless, another high octane display on the right wing. Photo: Jason Brown
10. Callum Lang - 7
Buzzed around as ever, trying to make things happen and put one wonderful ball through to Ritchie in the first half for a cracking chance. Never gave Burnley a moment’s peace. Photo: Jason Brown
11. Josh Murphy - 7
Produced some wonderful crosses from the left, particularly early in the second half, to ramp up the creativity following a disappointing Millwall evening. Always a threat and always so difficult to handle. Photo: Jason Brown
12. Colby Bishop - 6
Had a glorious chance to win it in stoppage time, but fired his right-footed shot over the bar. Improvement on his Millwall display, when he looked a little tired, yet he will be ruing that miss. Photo: Jason Brown