The Hawks started the game in second spot, only behind the Darts on goal difference, but an early 2nd minute goal proved decisive for the hosts.
Elliot Bradbrook's ball forward was nodded down by Alfie Pavey for Tom Murphy, who fired past Ryan Young at his near post.
Havant enjoyed plenty of possession but only forced Darts keeper Deren Ibrahim into serious action once during the whole 90 minutes.
Havant slipped to third after Hampton & Richmond Borough`s 2-1 win at lower mid-table Eastbourne Borough.
The Beavers had to come back from being an 8th minute goal down at Priory Lane with a fine individual effort from Lloyd Dawes.
But the visitors were back on level terms right on half-time when Charlie Wassmer finished from close range after his initial effort had been saved.
And what proved to be the winner arrived on the hour-mark with a near-post Max Kretzschmar header.
Braintree Town are fourth after another draw, this time 1-1 at Hardenhuish Park against lowly Chippenham Town.
A fourth draw in five games came after the Bluebirds had opened the scoring through Jake Jackson in the very first minute of the match.
However, the Iron pulled level 9 minutes later when a long ball forward was misjudged by Chippenham `keeper Jared Thompson and Reece Grant was able to pounce and tap the ball into an empty net.
Chelmsford City are sixth but were also only able to pick up a single point from a 1-1 draw at Melbourne Stadium against eleventh-placed Bath City.
It was the visiting Romans who broke the deadlock 2 minutes into the second half when Andy Watkins` initial effort was only half-saved by Sam Beasant and Ross Stearn slid in at the far post to poke in to end his long wait for a goal.
And that looked a though it was going to be enough until the Clarets popped up with an 88th minute leveller when a deep cross was headed into the bottom corner by Tom Hitchcock.
Chelmsford moved above Hemel Hempstead Town after they suffered a 2-1 defeat at Vauxhall Road to mid-table Hungerford Town.
This was another game that was goalless at the break, and then within 3 minutes of the second half getting underway, Jordan Parkes put the home side in front.
But 10 minutes later, Nick Bignall equalised and the Crusaders won it through Luke Hopper on 65 minutes.
The day`s other game saw bottom-markers Whitehawk win for the second time in three games and a third match unbeaten as they won 1-0 at another side in the bottom third, Oxford City.
What proved to be the only goal of the game arrived on 58 minutes when Gold Omotayo curled a free-kick past City`s debutant `keeper Craig King and into the corner of the net.