Cheltenham had the best effort of the first half with the wind in their favour but could not find a way past in-form County goalkeeper Saul Deeney who made a series of fine saves to keep the score at 0-0 going into half-time.
Deeney performed the first of his heroics after only 13 minutes when he dived full-length to palm aside a curling shot from a tight angle by Martin Devaney.
Irish duo Devaney and Grant McCann were at the centre of everything Cheltenham did creatively but each time they got close to goal either the finish was lacking or Deeney was in the way.
The Derry-born 21 year old touched a long-range shot from McCann over the bar then produced an outrageous display of reflexes to claw away a shot from point-blank range by Steve Guinan.
Cheltenham also wasted chances in the first half with David Bird shooting across the face of goal and Jamie Victory heading wide from a good position.
The deadlock was almost broken shortly before half-time when a colossal clearance from goalkeeper Shane Higgs almost bounced into the net but once again Deeney was there to save.
After the break Notts County, who had defended stoutly in difficult conditions during the first 45 minutes, got the ball down and played some slick counter-attacking football.
They took the lead after 58 minutes when leading scorer Hurst pounced on a loose ball in the penalty area to score from 12 yards – his 11th goal of the season.
The result was put beyond doubt 11 minutes later when a long throw by Matthew Gill was met by a weak punch from Higgs and Palmer hooked a shot back past him from the edge of the box.
After that Cheltenham were rarely in the game and County could have extended their lead when Palmer saw another powerful shot saved by Higgs, who also dived at the feet of Hurst late on after a poor back-pass from Shane Duff.
The win eased County's fears of being dragged into the relegation dogfight, while Cheltenham's hopes of challenging for the play-offs will continue to look forlorn until they can find someone to put the ball in the net on a regular basis.