The Huddersfield striker netted with a sublime chip midway through the first half as the Bees suffered their second successive defeat at a key stage of the season.
Brentford started the game in the top six but this defeat and results elsewhere mean that Sheffield Wednesday have replaced them in the play-off zone.
The home side threw everything into attack in the final stages of the match but they could not find a way though as Hudderfield held on to Abbott's first-half effort.
And what a goal it was as Brentford were made to pay for some sloppy play.
Abbott was given acres of space after a mix-up in the Brentford midfield and grabbed his opportunity to race unchallenged towards goal.
Goalkeeper Stuart Nelson raced off his line to narrow the gap, but the striker exquisitely chipped the ball over his head and into the net from 20 yards.
Up until that point Brentford had been poor, but the goal sparked them into life and they almost grabbed an equaliser before the break.
However, Paul Rachubka made a brilliant save to keep out Sam Sodje's flicked header from Kevin O'Conner's pin-point cross.
Brentford continued to dominate in the second half but former Manchester United keeper Rachubka was again equal to everything they produced.
He made a decent block with his legs from Jay Tabb's half-volley before reacting quickly to keep out Andy Frampton's 20-yard effort that took a wicked deflection off team-mate Darren Pratley.
Rachubka was also commanding his area well as he claimed cross after cross as Brentford increasingly resorted to high balls into the box.
Deon Burton fired into the side netting on more than one occasion from decent positions before Huddersfield wasted a glorious chance to ease their defensive nerves.
From a rare breakaway substitute Danny Schofield found space in the box, but he dragged his shot wide of the far post as Brentford survived.