This was the Clarets first defeat in seven games and puts Kenny Jackett's men within seven points of the top-six.
"It keeps the play-offs open to us and that is an achievement after coming up from League One," said a delighted Jackett.
"We have nothing to lose - we have to keep believing - but a high number of points is required.
"If we attack it like we did today, and with the same appetite, then we have to keep believing we have a chance.
"From our point of view now we want to make sure everybody stays fit and this type of week can galvanise you.
"Our away record recently hasn't been as good as it should be - but we haven't been able to get over that line."
He added: "It was a very pleasing performance. We managed to keep the crowd quiet which was important.
"Burnley are a very strong side - a side I rate highly - and once they start to play they can step up and put on very good performances."
Burnley counterpart Eddie Howe said the Clarets' inability to defend set-plays had proved costly.
He said: "It was a disappointing performance and we were below-par today.
"But credit to Millwall, they defended very well, got bodies behind the ball and made it difficult.
"Their gameplan worked well and we lacked our spark today - we gave the ball away too often.
"We conceded the goal in our best spell and it is a real disappointment to concede from a set play.
"It's back to the training ground to try and put that right.
"Sometimes you learn more from a defeat and we have learned a lot today.
"An important part of winning is how you deal with setbacks and we need to try and find the intensity that was missing today."