Macclesfield actually took the lead at Moss Road courtesy of debutant Marcus Marshall, before shipping three goals.
Elsewhere, Michael Smith netted a hat-trick as Accrington Stanley edged out Gillingham by the odd goal in seven.
Shrewsbury are up to fourth with a 2-0 win over Hereford, while Aldershot took all three points from their trip to AFC Wimbledon thanks to a 2-1 win.
Scott Doe scored the winner for Dagenham & Redbridge as they moved out the relegation zone with a 3-2 victory over Rotherham United.
The Daggers' place in the drop zone has been taken by Plymouth Argyle, who lost 1-0 to Port Vale.
League Two scores: AFC Wimbledon 1-2 Aldershot, Accrington 4-3 Gillingham, Barnet 2-0 Crewe, Bristol Rovers 2-1 Bradford, Dagenham 3-2 Rotherham, Hereford 0-2 Shrewsbury, Macclesfield 1-3 Cheltenham, Port Vale 1-0 Plymouth, Torquay 1-0 Northampton.