The 20-year-old fired in from six yards in the 79th minute to cancel out a 20th minute opener from Charlie Ide for the hosts - also his first strike of the campaign.
Terry Butcher's mid-table League Two outfit had looked like claiming a cup scalp as Luton struggled to break down their opponents, but Andrew's late act ensured both teams were left in the hat for the second round draw.
Brentford showed three changes to the team that lost 1-0 at Macclesfield in midweek with one of those being Lewis Emanuel, on loan from the Hatters, dropping from the side.
Luton welcomed back leading goalscorer Paul Furlong in place of Andrew after the 0-0 Tuesday night draw with League One leaders Carlisle.
The 39-year-old hitman had been missing with a hamstring injury, but his return was short-lived as he limped off just five minutes in to be replaced by Andrew.
In a sparsely filled Kenilworth Road, Ide and David Edwards both hit early shots off target for each team before Ricky Shakes drove an angled effort wide in the 13th minute after Lee Thorpe had caught Matthew Spring napping.
The first shot on target came three minutes later as David Bell beat Kevin O'Connor to a Darren Currie cross and drove in a left-footed effort that Simon Brown pushed aside.
But it was the Bees who opened the scoring in the 20th minute when Thorpe headed down a cross from the left and Ide showed great speed of thought to hook a 12-yard shot over the head of Dean Brill and into the net.
The Hatters had a glorious chance to get back on level terms seven minutes later when Spring shanked a speculative 35-yard effort, but the ball cannoned off Steve Robinson into the path of Edwards who, with the goal at his mercy eight yards out, somehow sidefooted wide.
Spring was then forced to leave the field to receive treatment for a facial injury and with Luton temporarily down to ten men Thorpe almost burst through on to a long ball in the 38th minute only for Chris Coyne to charge down his shot.
Chants of 'Blackwell out' rang around the ground as the players left the field at half-time and the Luton manager responded by bringing on Drew Talbot for Edwards at the interval.
It had little immediate effect though as Glenn Poole almost doubled Brentford's lead just 15 seconds after the restart with a 35-yard effort that fizzed just past the post.
To make matters worse for the hosts, the build-up to that opening saw on-loan Bolton defender Jaroslaw Fojut pick up a serious-looking injury that caused him to be stretchered from the field.
The changes at least saw some response from Luton as Bees keeper Brown was forced to save Robinson's 25-yarder, Currie's 30-yard free-kick and Spring's cross-shot all in the space of 12 minutes.
Currie then hit a swerving 35-yard effort just over the bar in the 65th minute after a terrier-like tackle from Robinson in the middle of the park had created the opportunity.
Poole tried his luck from the same distance for Brentford 60 seconds later, but Brill gathered at the second attempt.
Sammy Moore then fired just past the far post from 25 yards in the 73rd minute after turning Spring before Currie, Luton's liveliest performer, forced Brown into a save at his near post a minute later.
Currie was involved again as Luton grabbed the equaliser 11 minutes from time as the former Ipswich man's low cross from the right ricocheted into the path of Andrew, who made no mistake from four yards.