Edwards headed in the equaliser following a scramble in the penalty area and another fantastic goal from Noble put the Magpies in front at half-time after a nervy start to proceedings.
Noble added the third goal - his second - from the spot after former Crawley striker Izale McLeod had been fouled before Edwards nodded home his second of the night as Notts took advantage of their game in hand to move up the League Two table.
Notts manager Ricardo Moniz rang the changes following the late defeat at AFC Wimbledon on Saturday afternoon. Mike Edwards, Elliott Hewitt, Alan Smith and Liam Noble were named in the starting line-up with Thierry Audel, Genaro Snijders and Jon Stead dropping to the bench whilst Mawouna Amevor wasn't named in the matchday squad.
County threatened early on as Graham Burke sent a cross-field pass to Adam Campbell, who saw his shot deflect off Luke Rooney and away for a corner.
Noble's wicked cross looked like it was going to be turned in by McLeod, but Jon Ashton beat the striker to the ball and bundled it narrowly past his own post.
Crawley had a chance to take the lead when they broke quickly from the resulting corner but Stanley Aborah cut out the attack after a heavy touch by Lewis Young.
The visitors came forward again, Rooney dribbled into the area and forced Carroll into a good save with his legs at his near post.
At the other end, Noble sent a pass just out of McLeod's stride; the striker's goalbound prod lacking in power and accuracy.
Carroll was forced into another good stop when Murphy span away from Edwards inside the area and looked set to score, only for the veteran goalkeeper to smother the effort at close-range.
However, Murphy got his goal when Bobson Bawling charged down Carroll's clearance, leaving the Oldham loanee with a simple finish on an open goal.
McLeod had a header scrambled away from danger and Gill Swerts shot wide from distance but Notts were not kept waiting long for their equaliser.
Hewitt's long throw bounced around in the area and Edwards showed great determination to out-jump the pack to head across Freddie Woodman and into the far corner of the Kop goal.
Within a minute, Notts had completed a remarkable and quick-fire turnaround. Campbell's probing pass on the edge of the area found Noble, who cut inside his marker and curled into the top corner to the delight of the home support.
The in-form midfielder could have scored again from another Hewitt long throw which fell for him at the edge of the area but his powerful effort was palmed away by Woodman.
Edwards slashed wildly at a shot 20 yards out to send the ball spiralling over the bar whilst Rooney shot wide at the other end when unmarked 12 yards out.
As the game headed towards half-time, Noble tried to find McLeod with a floated pass but the striker headed over the bar.
Just a couple of minutes after the second half restarted, Notts were gifted an opportunity to extend their lead when McLeod was fouled in the area and referee Mark Brown pointed to the spot.
Noble stepped up to send the goalkeeper the wrong way and the ball into the bottom left corner to double the advantage and register his fifth goal of the campaign.
It was almost 4-1 within a minute later, but McLeod's deft lob - after showing superb skill to control a long ball over his shoulder - landed on the roof of the net.
Burke, on his supposedly weaker right foot, then smacked a 35-yard effort towards the top corner, only to see the shot superbly tipped wide by Woodman.
Notts had the ball in the net for a fourth time just before the hour mark when Smith nodded Campbell's cross to the back post where McLeod side-footed into the top corner but the joy was short-lived as the referee's assistant had flagged for offside.
Crawley remained a threat and it took a one-handed save by Carroll to deny Sonny Bradley, who looked certain to head in from Rooney's free-kick.
Burke curled over from a free-kick on the angle of the penalty area and McLeod was denied by the visiting goalkeeper as he tried to slide the ball through his legs.
The fourth goal did arrive as Edwards nodded in Noble's corner at the near post for his third goal of the season with a quarter of an hour remaining.
Conor Henderson's free-kick took a nick off the wall to skip behind and Bradley headed off-target from the resulting corner but Notts were able to see the game out for a third successive home victory which moved them up to 16th in the table and just three points away from the play-off places.
Notts: Carroll; Hewitt, Edwards, Hollis, Swerts; Smith, Aborah; Campbell, Noble, Burke (Jenner, 69); McLeod (Stead, 78).
Subs not used: Loach, Audel, Sprockel, Thompson, Snijders.
Attendance: 3,267 (27 away)