Yoane Wissa's two first-half strikes sent the visitors into half time two goals to the good, with the Bees topping off a fantastic afternoon with a further three after the break.
The result leaves Brentford looking extremely likely to stay in the Premier League for a fourth successive season, taking them 10 points clear of the drop zone.
On the opposing end, the heavy defeat deals a potentially crushing blow to the Hatters' survival efforts with this being deemed as one of the biggest opportunities to pick up points in their remaining fixtures.
Relegation rivals Burnley's victory at Bramall Lane leaves them just a point behind Luton, while Nottingham Forest and Everton, who are one and two points better off respectively than the Hatters, are set to face off on Sunday.
The first half saw Rob Edwards's men dominate the possession, holding 57% of the ball, but the best chances largely went Brentford's way.
Keane Lewis-Potter should have opened the scoring in the eighth minute when he was played through into the Luton box, but the forward sliced through the ball on his weaker left foot, missing the target altogether.
Just minutes later Luton had their best chance of the first half, when Tahith Chong's effort was deflected back into the path of Alfie Doughty, with his fiercely-struck shot stinging the gloves of Mark Flekken.
The visitors found their rhythm after that chance, giving the hosts very little space to create any real opportunities while also managing to continually exploit the Hatters on the break.
Sergio Reguilon's dipping cross was met by Lewis-Potter once again, but his headed shot was unbelievably saved by Thomas Kaminski, with the Belgian clawing the strike wide of his near post.
The goalkeeper's heroics meant little though as four minutes later Wissa opened the scoring with a fantastic strike into the top-right corner having been played through by Bryan Mbeumo.
The chances dried up in the following 20 minutes with both sides able to cancel out any attacks from coming together.
However, the visitors upped a gear in the final minutes of the half as Mbeumo unluckily hit the bar with his curling shot before turning creator as he assisted Wissa's second of the game.
The Cameroonian went past his man and squared the ball across the box, initially deflecting up off Reece Burke with the defender attempting a flicked clearance.
The attempted clearance was poor though as the ball landed at Wissa's feet, with the striker bundling in to claim his 10th goal of the season.
The same pattern emerged in the second half as Luton saw most of the ball, but Brentford were the much more dangerous of the two teams.
Ethan Pinnock scored the third just after the hour mark as Reguilon's outswinging corner was misjudged by Carlton Morris, catching Gabriel Osho off guard with Pinnock's unchallenged header flying into the top corner.
Moments later the scoring continued, this time through Lewis-Potter who made up for his two first-half misses as he headed in Mbeumo's looped cross to make it four.
The cherry on top came for Brentford in the 85th minute thanks to some brilliant work once again by Mbeumo, who skipped past his man before playing in Vitaly Janelt, who pulled the ball back to Kevin Schade for an easy finish.
Luton did manage to at least get one back, scoring in the 92nd minute through Luke Berry after the midfielder latched onto Janelt's loose pass before rounding the keeper and finishing into an empty net.