Suicidal defending at both ends contributed to an entertaining Saturday that could have seen either team finish winners, with the most frenzied action unfolding during a roller-coaster second half.
Norwich pegged back Michu's opener with goals from Robert Snodgrass and Michael Turner that appeared to have secured three important points which would have eased their relegation concerns.
But Moore had the final say 15 minutes from time, denying the Canaries only their third Premier League victory since December.
Substitute Norwich forward Grant Holt was lucky to escape punishment for a nasty challenge on Chico Flores that him stamp down on the defender's shin.
Swansea have struggled since winning the Capital One Cup and avoided fourth defeat in five games since triumphing over Bradford at Wembley, but will be disappointed by Michu's wastefulness.
The Spaniard carved Norwich open in the ninth minute but Moore dragged his shot wide after making an intelligent run that enabled him to gather the defence-splitting pass.
Michu drove the ball into the net but Swansea were infuriated that referee Michael Oliver had spotted a tame-looking foul on Jonny Howson in the build-up.
Goalkeeper Mark Bunn, back after serving a one-match ban, lasted only 15 minutes after appearing to injure his groin during a clearance, so Lee Camp stepped off the bench.
Elliott Bennett took a theatrical dive but Ben Davies had clearly pulled him around the neck and was booked as a result, giving Norwich an inviting free-kick that Robert Snodgrass blasted at the wall.
Showing persistence, however, the Scot picked up the rebound and fought his way to the byline before passing across the face of goal.
Only a fine diving save from Michel Vorm denied Turner from close range, but had the shot been struck just a little harder the ball would have evaded the Holland keeper.
Touches of class from Jonathan de Guzman and Michu saw the visitors take a deserved lead in the 35th minute.
De Guzman weaved a path to the byline and then squeezed a pass between two Norwich defenders to the feet of Michu who rifled the ball into the roof of the net with his first touch.
Nathan Dyer could have made it 2-0 a moment later and Swansea lamented the miss chance in the 40th minute when Norwich equalised.
The visitors were caught on the counter with Elliott Bennett delivering a pinpoint cross to the far post where Snodgrass produced a successful diving header.
An audacious lob from Wayne Routledge put Moore one on one with Camp, but the replacement responded with a marvellous save.
Norwich had grown in confidence since Snodgrass' equaliser and with a little help from some weak defending they took the lead 15 minutes after the interval.
A free-kick was despatched to Turner at the far post and the defender, who was unmarked after being allowed to roam free, had the time and space to drive the ball home.
Michu blundered with a magnificent chance to equalise by directing his attempt straight at Camp, but Norwich were equally culpable for failing to capitalise on an opportunity when Kei Kamara was expertly dispossessed by Ashley Williams after delaying his shot.
With both teams struggling in defence, it was only a matter of time before the next goal arrived and this time it was delivered by Swansea.
Angel Rangel's cross was headed into the path of an unmarked Moore, who buried the chance to grab the point and deny the home side of a much-needed win.