The Bees' victory, coupled with a 1-0 defeat for rivals Rotherham United at Sheffield United, means that Mark Warburton's men are now nine points clear of third place in the League One table with five games remaining.
However, Jones made a poor error after the break, spilling Jonathan Douglas's header into the net to gift the Bees a goal that proved to be the winner.
The game ended on a controversial note when a fan forcefully threw the ball at Crawley full-back Paul Connolly, who was waiting to take a throw-in.
Connolly then squared up to the fan and swung his right arm at him, before being ushered away by stewards. He was not shown a card by referee Steve Martin.