Swansea threatened first when 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.
Canaries goalkeeper Mark Bunn, back after suspension, lasted only 15 minutes after appearing to injure his groin during a clearance, so Lee Camp stepped off the bench.
Norwich defender Michael Turner had been denied before 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 the home side equalised.
The visitors were caught on the counter with Elliott Bennett delivering a pinpoint cross to the far post, where Robert 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 after Snodgrass' equaliser and with a little help from some weak defending they took the lead on the hour. A free-kick was despatched to Turner at the far post and the defender, unmarked after being allowed to roam free, had the time and space to drive the ball home.
However, the visitors levelled 15 minutes from time when Angel Rangel's cross was headed into the path of an unmarked Moore, who buried the chance.