However, the Lions should have claimed all three after they spurned a host of chances following Clarke Carlisle's seventh-minute red card.
Carlisle was dismissed for a rash challenge on Theo Robinson and, although Millwall had several opportunities to open the scoring, it was Burnley who took the lead thanks to Jay Rodriguez's header.
It was the Clarets' first effort on goal but Kenny Jackett's men soon levelled thanks to Barron's low effort.
Although Alan Dunne hit the post in the second-half, Millwall could not take advantage of the extra man.
Following a cagey opening in which both teams were heavy in the tackle, Burnley's afternoon was made a lot tougher when defender Carlisle was given a straight red card.
The veteran centre-half made a rash lunge on Millwall debutant Robinson and, although his dismissal was a little harsh, it left the Clarets with a mountain to climb.
Millwall, who halted a run of three consecutive defeats when they secured a midweek draw against Championship leaders QPR, predictably laid siege to the visitors' goal.
Steve Morison and Robinson both forced fine saves from Burnley goalkeeper Brian Jensen, while Dunne narrowly missed the target from just outside the area.
Burnley's fortunes went from bad to worse as Andre Bikey, on his 100th career league appearance, went off injured and was replaced by Leon Cort.
However, against the run of play, Rodriguez gave Burnley the lead in the 22nd minute when he rose highest from Chris Eagles' right-wing cross and headed past Millwall goalkeeper David Forde.
Barron deservedly levelled for the hosts four minutes later, though, when he drilled the ball home from the edge of the area following Robinson's knockdown.
After the equaliser, Millwall failed to seize the initiative though in first-half injury time, Dunne's header forced a fine save from Jensen.
After the interval, Chris Hackett's fine volley nearly fired Millwall into the lead for the first time, but his effort went narrowly wide.
The midfielder's corner then set up Dunne and although this time the 28-year-old's header left Jensen stranded, he was denied his first league goal by the post.
As in the first half Millwall were not able to make their superiority count, but kept pushing forward and Burnley were again indebted to Jensen when he denied Neil Harris from 12 yards.
Substitute Danny Schofield's point-blank strike was then magnificently tipped over the bar by Jensen, while Dunne twice headed over late on.