Jason Scotland capitalised on a slow start by the home side to send Swansea in a goal to the good at half-time.
But Millwall hit back through Zak Whitbread and Alan Dunne, who converted from the spot, before Anderson rescued a share of the spoils for Roberto Martinez's visitors.
The build-up to the game was dominated by news of the appointment of Kenny Jackett, the Manchester City reserve-team boss, as Millwall manager following the sacking of Willie Donachie a month ago.
Jackett, who relinquished his post as Swansea boss by mutual consent in February, was not scheduled to take over until the day after the match and did not attend.
Had he done so, he would have seen an injury-ravaged Millwall side overcome a disjointed first-half performance despite being deprived by injury of a host of first-team regulars including Dave Brammer, Danny Senda, Neil Harris and Ryan Smith.
Millwall enjoyed plenty of possession throughout, but frequently struggled to cope with Swansea's swift and intelligent counter-attacking.
Their best chance of the first half came when Ali Fuseini forced a save of superb athleticism from Dorus de Vries, the Swansea goalkeeper recently nominated for the League One player-of-the-month award.
By then, however, the visitors had snatched the lead courtesy of a swift counter-attacking raid led by the lively Anderson.
With 18 minutes gone, Anderson raced clear of his marker to unleash a stinging shot that Lenny Pidgeley could only parry into the path of Scotland, who rifled the ball home for his eighth goal of the season.
However, Millwall struck back swiftly after the interval with substitute Whitbread heading home Jamie O'Hara's 58th minute corner to lift the Lions' spirits, before Dennis Lawrence handled an Alexander cross to concede a 69th minute penalty which was expertly converted by Dunne.
For a time it looked as though Millwall might snatch all three points, but 12 minutes from time they failed to clear a Scotland cross, sparking chaos in defence and enabling Anderson to fire Swansea level.
De Vries was forced to save with his feet from Fuseini's long-range piledriver in the closing minutes, while Alexander flashed a header wide in injury time.