Goals from Scott McDonald and Danny Shittu put the Lions two goals to the good but the Clarets hit back through Sam Vokes and Lowry's second-half own goal.
It is a third straight draw for Steve Lomas' side and it means Burnley were unable to make it eight consecutive league wins, failing to keep up with their best ever start to a season, back in 1897.
Millwall began the brighter and had the first genuine chance of the match through Nicky Bailey, who saw his 20-yard effort well saved by Burnley goalkeeper Tom Heaton.
Both sides were creating opportunities with Martyn Waghorn testing Heaton at the near post before Burnley's in-form striker Danny Ings' shot was well blocked by Paul Robinson.
But on 23 minutes Millwall had a shock lead through Australian forward McDonald, who volleyed into the far corner from eight yards after Robinson's accurate knock-down.
It was only the second time Burnley had conceded the first goal of a game this season with the other being their only loss, which came away against Brighton.
But they were 2-0 down 15 minutes later, through an unexpected source, when Lions centre-back Shittu netted with a deflected long-range volley which sailed past a wrong-footed Heaton.
Just a minute later however, the Clarets halved the deficit through Vokes.
Kieran Trippier's cross was dealt with poorly by the Millwall defence and the ball fell to Vokes, who rose highest to nod home his ninth goal of the season.
Burnley pressed for a leveller before the break but to no avail as Millwall held out for a 2-1 lead at the interval.
Again the hosts were livelier at the start of the second half with Waghorn twice going close with shots from distance.
But on 55 minutes, Michael Kightly found space in the final third for the away side and his dragged shot, which was heading wide, deflected in off Lowry and in as he attempted to clear.
That gave the visitors the impetus and Scott Arfield looked to have given the Clarets the lead on 66 minutes but his right-footed volley crashed off the inside of the post and away.
Arfield then went close with a header for Burnley while a David Jones cross may have caused trouble for Millwall but no-one could find the finishing touch for the away side.
Burnley continued to search for the winner but Millwall, who have not won in the league since the end of September, were happy to settle for a point - and for all of the visitors' endeavour, that is exactly what they earned.