Debutant Jordan Kirkpatrick put Clyde in front after eight minutes but Arbroath turned the game on its head early in the second half.
Bobby Linn's penalty levelled 11 minutes after the break, then Richard Little's right-footed effort two minutes later left Clyde stunned.
Clyde failed to capitalise on leaders East Fife's 1-1 draw with Queen's Park.
Struggling Arbroath came from behind to earn a victory which denied Clyde the opportunity to go top of League Two.