Aylesbury started as they were to go on by dominating posession, and as early as the fifth minute they could have taken the lead. Jonathan Bewers' shot was diverted by Neil Lazarus but fell wide of the post.
Edgeworth also saw an effort deflected narrowly wide, before Peat had to be alert to palm away Gavin Jaggard's effort after the striker played a neat one-two with Edgeworth.
The Ducks suffered a blow after nineteen minutes when full-back Dean Powles was forced off injured, to be replaced by Greg Williams.
Cheshunt sprung a surprise seconds later when they nearly scored from a corner, but Rashid did excellently to save on his line. At the other end Neil Lazarus blazed a good chance over the bar.
Substitute Williams was a revelation after coming on, his no nonsense style unsettling Cheshunt, and he had a strong case for a penalty on half an hour when he burst into the box and went down under challenge, but referee Mr Farries wasn't interested.
The Ambers seemed intent on using their pace to attack United, but on numerous occasions were caught offside by Aylesbury's rearguard. One chance saw George Gregoriou escape though, but Rob Gould recovered superbly to catch up and make a saving tackle inside the area.
Three minutes before the interval Edgeworth had the best chance of the game. The ball fell to his feet, but Peat threw himself at the shot and managed to divert the effort wide.
Lloyd Opara had the final action of the half, capping his hapless first half with a wayward shot well over the bar.
Half-time: Aylesbury 0-0 Cheshunt
The second half began with Matt Hayward sending a free header wide from a corner, before Cheshunt asserted some real pressure for the first time. Gregoriou came forward but saw his shot held by Rashid, before Opara fired wide when well placed.
Gregoriou missed the target again two minutes later, after Rashid had parried an initial effort, before the Aylesbury stopper got down well to smother Gregoriou's shot.
The forward then summed up a frustrating night for Cheshunt's strikers when he spurned their best opportunity of the match. The pacey Clark broke down the left, cut the ball back for Gregoriou, but he directed his shot high over the crossbar.
After an hours play Danny Nicholls made a substitution which truly inspired his side into life, replacing Lazarus with youngster Aston Goss.
Suddenly United were back in command, inventive in their play for the first time since the interval. Jaggard went close when he curled a well struck effort just wide.
Six minutes after coming on, Goss was at the forefront of the move that gave Aylesbury the lead. Hayward played the winger in down the left, he squared for Roni Joe who turned at the near post and picked out Scott McCafferty who calmly slotted home. 1-0 Aylesbury
Goss went close himself, feeding on Joe's ball into his feet before shooting wide on the turn. Jaggard then tried an ambitious long-range effort which flew narrowly past the upright.
United continued to fashion chances, but a second goal just wasn't forthcoming. Gould unleashed a fierce left-footed shot wide, and Jaggard shot straight at Peat having been put clean through.
Further efforts came from McCafferty and Bewers, and Jaggard saw an effort blocked. Roni Joe had the best of the remaining chances though, having broken free he fired in low but Peat got down to tip the ball wide.
Rob Gould headed a corner wide late on, as Aylesbury had to settle for a solitary goal, but they won't mind too much with the three points finally recorded.