The Mariners had left with just five minutes on the clock after Tom Broadbent headed his side into the lead.
However, the visitors stormed back after the break and turned the game around thanks to goals from Paul Ennis and Mike Potts.
That result means their lead at the top of the division now stands at 10 points after Radcliffe climbed up to second with a 2-0 win away at Marske United.
Despite goalkeeper Jack McIntyre being sent off 16 minutes from time, forcing outfield player Rick Smith to play in net during the closing stages, late goals from Olly Thornley and Jake Thompson ensured the Boro returned to Greater Manchester with all three points.
The night’s two other matches - Bamber Bridge against Morpeth Town and Marine’s trip to Matlock Town - were both postponed following pitch inspections.
New dates for those fixtures will be announced in due course.
Results - Tuesday 14 March
Marske United 0-2 Radcliffe
South Shields 1-2 FC United of Manchester
Photo credit: Radcliffe
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