Central defender Bird was punished for two bookable offences in the space of five minutes at the end of the first half, and it proved to be a turning point in the match as goals from Jake Robinson and Jai Reason settled the game in the second half.
The first occurred high up the pitch as Billericay attempted to break away following a Slough attack, and was controversial in that Bird looked to have won the ball cleanly. However, worse was to follow when Billy Knott went to ground under a challenge from the defender just outside the penalty area.
Slough were left aggrieved at the inconsistency of refereeing decisions, with Billericay’s Joe Felix – also on a yellow card – going unpunished for committing another offence toward the end of the half.
Both sides had chances in the first half, with the Rebels finding Billericay goalkeeper Dion-Curtis Henry on top form to deny Malachi Linton in particular on a number of occasions, the on-loan Wycombe Wanderers striker making his first start.
Henry’s first good stop came after just eight minutes, as he kept out a goal-bound effort from Warren Harris.
Billericay talisman Robinson saw a shot deflected wide thanks to a big tackle from Slough skipper Sam Togwell - on his 100th appearance for the club - and shortly after, Billy Knott fired narrowly wide from outside the penalty area.
Linton threatened the hosts’ goal twice in three minutes, first having a shot deflected wide after a strong run into the box, and then forcing Henry into another fine save following another driving run.
Slough racked up numerous corner kicks but were unable to make their pressure count, and Bird’s dismissal set up a challenging second half.
Lee Togwell was drafted in after the red card to partner his brother Sam at the heart of the defence, with Slough sacrificing attacker Paul Hodges - but he barely had time to settle into the match before the hosts took the lead.
Jake Robinson struck inside the opening minute of the second half, punishing indecision in the Slough defence and firing into the bottom corner from just inside the penalty area.
In response, Cam McGilp fired a right-footed shot over the bar from 25 yards, and Linton and Josh Jackman both had efforts blocked as Slough continued to push forward.
A big chance arrived on 65 minutes, but Linton could only head over the bar from inside the six yard box from Sam Togwell’s header back across goal.
Goalkeeper Henry came to Billericay’s rescue yet again eight minutes later, making another save to deny Linton as Slough introduced another striker in Dan Roberts.
And Slough were punished in spectacular style six minutes from time when Jai Reason smashed the ball into the top corner from 25 yards after a free kick was touched into his path.
Max Worsfold had two late half-chances to pull a goal back, but his first effort was blocked and his follow-up well saved by Henry.
Billericay Town Line up
Dion-Curtis Henry, Chevron McLean, Joe Felix (Jai Reason HT), Ronnie Henry, Michael Chambers, Josh Hutchinson (Toyosi Olusanya 66), Billy Knott, Jake Robinson, Shamir Fenelon (Lewis Knight 87), Lawrie Wilson, Luke Wanadio. Unused substitutes: Kyel Reid, Ruben Soares Jr.