Chris Hughton's side came from behind to take the lead through Robert Snodgrass and Michael Turner but the Swans hit back late on to increase the Norfolk club's fear of the drop
Michu converted Jonathan de Guzman's cut-back to give the Swans a deserved lead but despite their continued dominance of possession, the home side managed hit back either side of half-time.
Elliott Bennett's cross was headed in by Robert Snodgrass five minutes after Michu's opener before another Bennett delivery was tucked away by Michael Turner.
Moore bundled home with 15 minutes to play, however, after De Guzman had headed Dwight Tiendalli's cross into the six-yard box to ensure the spoils were shared.
Kei Kamara led the line for Norwich ahead of Grant Holt, who was on the bench, while Alexander Tettey remained among the substitutes as he continues his comeback from a knee injury. Meanwhile, Moore started up front for Swansea with Michu offering support from a deeper position.
The Spaniard threatened to exploit the gap between the hosts' defence and midfield from the off, tamely shooting wide from 20 yards before finding the bottom corner from a similar position only to see referee Michael Oliver call the play back for a foul by Moore on Jonny Howson.
Norwich did not test Michel Vorm until past the 20-minute mark after opening the match in familiarly uninspiring style in attack. The Dutch goalkeeper did brilliantly when called into action, though, to palm away Turner's effort after Snodgrass' run and cross from the right.
First-half chances were otherwise few and far between but the visitors always looked more likely to manufacture the sort of cutting move that resulted in the contest's opening goal. After a Swansea corner, De Guzman smartly turned onto Chico Flores' pass and cut the ball back for Michu, who finished expertly into the roof of the net from 12 yards.
That led to more than a little dissent in the stands, with the threat of the relegation battle increasingly looming over the Canaries, but before the half was out they were level. Seconds after Nathan Dyer had shot narrowly wide at the other end, Bennett's cross to the back post was nodded in by Snodgrass.
Norwich's mini-revival did not pick up where it left off after the break, however, and Lee Camp, who had replaced the injured Mark Bunn in the first half, needed to race quickly off his line to deny Moore after Wayne Routledge's has cleverly chipped his pass over the head of Sebastien Bassong.
Michael Laudrup's team's failure to make the most of their dominance would be punished once again. For the second time, Bennett was the provider, this time whipping in a free kick from the right that was swept in by Turner, who was lurking behind the Swansea defence.
Norwich appeared to be fairly comfortable thereafter, one free header wasted by Michu aside, but Moore capitalised on De Guzman's smart header into the six-yard box to slot past Camp and increase the pressure on Hughton.