Brighton may have started the afternoon ten places and 11 points better off than their hosts, but on an afternoon when Alexander was at his exuberant best, Lions against Seagulls proved no contest.
The Millwall striker opened the scoring on six minutes, doubled his tally with a thumping header nine minutes after the break and secured ownership of the match ball with a cool penalty 20 minutes from time.
Yet, for all Millwall's eventual dominance and despite the ultimately emphatic scoreline, it was Brighton who had started a typically hell-for-leather League One clash the stronger.
The initial signs were good for the visitors, who won two early corners and forced Lenny Pidgeley, the Millwall goalkeeper, into a fingertip save from Jake Robinson's curling effort.
But their momentum was checked on six minutes when Dave Brammer's well-flighted free-kick, delivered from a position deep on the left wing, created the simplest of chances for the unmarked Alexander, who nodded home at the far post.
It was an uncharacteristic lapse from a defence that had previously conceded only eight times away from home, but Brighton were soon pouring forward again, skipper Dean Hammond rallying the troops as Dean Wilkins' side pressed for an equaliser.
That they were denied is a tribute to a well-organised Millwall defence that, although breached six times in two games coming into this encounter, looked balanced and secure throughout.
Central to this new-found solidity was the decision of manager Kenny Jackett to shift right-sided defender Danny Senda into unfamiliar territory at left-back.
Allied with the experience and defensive nous of Marcus Bignot, the experienced right-back on loan from QPR, Millwall had solid foundations on which to build their attacking raids.
Shortly after the interval Bignot proved that he is no slouch going forward either, splitting the Brighton defence with a precise cross for Alexander to slam home his second.
By then Wilkins had thrown on midfielder Dean Cox, who was recently omitted from the side for a breach of club discipline. But when Michel Kuipers, the hapless Brighton goalkeeper, felled Alexander to concede a 69th minute penalty, the writing was on the wall.
Alexander thumped home the spot-kick to complete his three-goal salvo, giving Millwall their third win in six league games and leaving Brighton to rue missed first-half opportunities.