Although City travelled south down in seventeenth place in the table, they had only lost once in their last ten league games and had won at Hungerford Town last Saturday.
And they twice came from behind here before eventually falling to a best-of-five-goal defeat.
The Hawks took a 9th minute lead with a Matt Tubbs effort but City ended the half strongly with December`s player of the month, Rob Sinclair, coming close to levelling the scores at the break.
Godfrey Poku’s speculative effort on 51 minutes found the bottom corner to level things up, but Havant were again in front just 3 minutes later with a header from Theo Lewis.
City again levelled on the hour-mark when top marksman Matt Paterson fired home from the penalty spot.
The lead lasted until the 78th minute when Tubbs stroked home his second of the night.
There was to be late drama with Reece Fleet shown a straight red card after a rather innocuous-looking incident.
The win wasn`t enough to take the Hawks above Dartford, who still lead courtesy of having a two-goal advantage.