Matt Tubbs' sixth goal of the season came from the spot on 70 minutes for Ryan Astles' challenge on Yemi Odubade.
Chester, who had beaten Aldershot 8-2 in midweek, were completely overrun by an Eastleigh side who had lost five of their previous seven league matches.
Visiting keeper Tony Thompson denied Luke Coulson, Josh Payne and Odubade before Tubbs' deserved winner.
The fight to reach the play-offs remains tense, with Eastleigh now only two points behind fifth-placed Braintree, while Chester are only eight points clear of trouble.
Chester manager Steve Burr told BBC Radio Merseyside:
Media playback is not supported on this device
"There were two things we felt hard done by. We had a goal disallowed in the first half when Ryan Higgins cut the ball back to Ross Hannah.
"And the lads are unhappy with the penalty decision. There was minimal contact. It was harsh, to say the least
"They have a very expensively assembled side with experience, who you could see are used to playing on that pitch and, in the end, it was settled by a penalty
"It was always going to be tough after what has happened this week, then the long journey down. We set off at seven o'clock this morning and you could see one or two tired legs out there. But I don't want to make excuses. We didn't create enough chances."
Eastleigh secured back-to-back wins for the first time since December as they narrowly defeated Chester.