The visitors exploded into action after the break with Liam Tomsett firing home two superb strikes but substitute Crawford topped them both as he chested the ball and volleyed it home in the dying stages.
The Bees made an excellent start to the match and with just five minutes played they took the lead through Marsh-Brown.
Curtis Weston slotted a great through ball into the forward who took it in his stride before firing past David Carnell.
Two minutes the Bees almost doubled their lead when Roberto Casabella saw his low shot saved by Carnell.
With 20 minutes gone it was Graham Stack’s turn to spring into action when he comfortably saved an Adam Griffin free-kick.
The Bees continued to dominate and they had further chances to increase their lead before the half hour mark.
Weston was first to be denied as he saw his shot somehow cleared off the line before Marsh-Brown sent a thunderous strike inches wide.
Barnet supporters will be pleased to know that on 27 minutes Luisma Villa returned from injury, however he replaced Luke Gambin who picked up an injury of his own. Further details on Gambin’s injury will be available when we have them.
After entering the fray Villa made an instant impact, his first attempt came with his first touch but it was blocked before he fired just wide from the follow up.
The Bees were desperate to add a second and with five minutes of the half remaining they did as Byrne smashed the ball home after Jake Hyde was denied from close range.
The away side rarely troubled the Barnet defence in the first-half but they pulled one back at the start of the second when Tomsett sent a stunner past Stack.
The Bees reacted positively to conceding and moments after doing so Byrne curled a free-kick just wide from 30-yards.
Barnet continued to push for a third to reclaim their two goal advantage but after top scorer Hyde missed a superb chance to make 3-1 the away side drew level as Tomsett again beat Stack from range.
The lead was blown and the points looked like they could disappear altogether as Hyde continued to push on before Crawford replaced Villa.
The forward’s first touch saw him chest the ball under his control before he volleyed the ball past Carnell to secure the win.