Striker David Brown netted the opener from the spot for Barrow after Bees keeper Ben Hamer was red carded for bringing down Jason Walker in the penalty box after 36 minutes.
And despite Charlie MacDonald grabbing one back for the League Two side, Henney crashed home Paul Brown’s cross from close range in the 71st minute to seal a famous win.
After a scrappy start the game changed on a moment of madness from Hamer when he hauled down Walker after failing to gather Steve McNulty’s through ball in slippery conditions.
Referee Colin Webster had no hesitation in pointing to the spot and sent Hamer for an early bath as David Brown kept his nerve to stroke home the spot kick.
Brentford equalised shortly into the second half when Sam Wood’s deflected cross found MacDonald and he slid the ball past keeper Alan Martin, but Barrow made their numerical advantage count when Paul Brown’s cross was flicked on by namesake David at the near post and Matt Henney nudged the ball over the line.