Wrexham were seeking to remove their League Cup hangover after losing at home to Welsh rivals Cardiff City on Tuesday night.
Coventry, the only unbeaten side in the English Football League, meantime, were closing in on a club-record seventh straight league win when Ephron Mason-Clark superbly slotted the Sky Blues ahead midway through the first half on Friday at the Racecourse Ground.
The omens were not good for the hosts given Frank Lampard's side had scored 34 goals in 12 games before arriving in north Wales and the Red Dragons had won only one of their previous seven.
But Moore turned the contest on its head with two goals in the space of nine second-half minutes, the Wales striker capitalising on a couple of excellent Josh Windass assists.
Moore then took his season tally to nine - from another Windass assist - seven minutes from time. His fine left-footed strike added to his earlier right-foot finish and header completed a first hat-trick for over seven years.
"I'm very pleased, it's a great night for us," Wrexham manager Phil Parkinson said on Sky Sports.
"That was a proper Wrexham performance, which our fans have been used to over the years.
" ... if ever there was one for us to gain belief as a group it was that one, but after tonight we've got to go up a level."
Coventry made the final moments interesting as Tatsuhiro Sakamoto gave them hope from 20 yards, but the home side held on to move up to 11th in the standings, on 17 points and 11 off top spot.
"I smelt it at halftime, even at 1-0 up ... we just dropped a bit," Lampard told Sky.
"Maybe that happens when you are 12 unbeaten and win six on the bounce.
"We can't believe our own hype. This league is long. And don't worry, we are good. But this is what happens to normal teams.
"So we better show in the next games what we really are again."
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