Carlisle hit back with a first-half penalty from Ian Harte but missed their chances and were made to pay by a pacy Bees side who were deadly on the break.
Former Notts County forward Weston did not have long to wait to make his mark, grabbing a 12th-minute lead for the visitors. Carlisle’s defence was all at sea as they tried to deal with Sam Saunders’ inswinging right-wing corner, and Weston was left all alone at the far post to head home from two yards out.
The Cumbrians were level just three minutes later when Harte crashed home a penalty after Saunders, back helping his defence, handled Graham Kavanagh’s cross from the left. In first-half added time Joe Anyinsah saw a close-range header hit the crossbar and bounce to safety.
Weston’s second in the 47th minute owed a lot to an error by Carlisle’s former Chelsea goalkeeper Lennie Pidgeley. The defence failed to deal with a low ball from the right and Weston turned to hit a shot from the edge of the box which went straight through Pidgeley’s legs and into the net.
Wood made it 3-1 in the 53rd minute when his skidding 25-yard drive took a big deflection off defender Richard Keogh and sent Pidgeley the wrong way before going in off the post. Weston could have had a hat-trick late in the second half but spoiled a 50-yard run by dragging his shot across the face of goal, then Saunders was robbed by Pidgeley’s out-stretched hand as he tried to round the keeper after another lightning break.