Newry need Donegal Celtic to lose their final two games to have any chance of avoiding the Premiership bottom spot and possible relegation.
It follows this defeat at the Showgrounds, courtesy of a Ryan Berry goal.
City's Peter McCann was denied twice by goal-line clearances in the first half while Newry keeper Andy Coleman denied Mark Surgenor.
Berry hit a superb half-volley winner a minute into the second half.
Newry were first to threaten with Andy Graham's glacing shot drifting just wide.
Lee Colligan and Denver Gage were the United players to deny McCann with goalline clearances.
The Sky Blues dominated the second half and Eamonn Murray, Colligan and Gary McCutcheon squandered chances.
Newry trail Donegal Celtic by six points with just two games left, although City boast a superior goal difference.