It was the first time the Reds had kept a cleansheet at Broadfield Stadium this season, thanks to goalkeeper Andy Little and a robust defence.
Reds boss Billy Smith has guided his side from two points to 18 since taking over at the end of October.
He said: "I thought we just about deserved it.
"We got a bit tired after shutting down their players to stop them from playing.
"There were some good performances out there. The goalie played very well. He made some excellent saves and Gareth Graham played well too.
"John Ugbah was back to his old self at the back, Francis Vines was a handful up front. Things are going in the right direction."
Confidence
Crawley have risen up to sixth from bottom with a new found confidence.
The hosts impressed in the first half while limiting Weymouth to a handful of chances.
Crawley scored on the half-hour when Vines was hauled down by Lee Bradford in the box and Payne struck home the resulting spot-kick.
But Little had to make a spectacular save from a looping Steve Tate header before the interval.
Weymouth improved after the break and Little made two more great saves.
The best was a reflex tip over to deny Tate again after the goalkeeper was forced to change direction quickly.
But Crawley battled away for three more points to boost morale before a trip to Gloucester on Saturday.
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