A goal at the start of the second half from the Blue Square South's top scorer Ollie Palmer edged City to the brink of relegation.
It was the big striker's 21st goal of the season and condemned City to their 22nd league defeat and it is only a case of when, not if, their relegation will be confirmed.
City made one change from the team that lost to leaders Welling with the fit again Marcus Martin replacing the injured Martin Watts.
In the first half City looked pretty solid against the in-form Hampshire side though the home side did hit the woodwork and the visitors did have a couple of half chances.
But they were caught cold after the restart when they failed to clear a free kick and Palmer pounced to score what was to prove the match winner.
City tried to come back and new signing Jordan Copp was unlucky not to equalise first being denied by the home keeper and then having his follow up cleared off the line at the expense of a corner.
Manager Lee Hodges brought on Matt Wright, Joe Broad and Stewart Yetton late on but City could not find an equaliser.
Ciity are now a massive 20 points away from safety with seven games left and its just a matter of time before their fate is confirmed.
