The Carrow Road club edged a point further away from the drop zone courtesy of the draw but could be pulled back to within two points depending on Sunday's results.
Michu had given Swansea the lead only for Norwich to come storming back either side of half-time through Robert Snodgrass and Michael Turner.
But Luke Moore then equalised for Swansea to ensure Chris Hughton's Norwich have now won just once in 15 Premier League games.
The game started badly for Norwich, with goalkeeper Mark Bunn, back after serving a one-match ban, lasting only 15 minutes after appearing to injure his groin, so Lee Camp stepped off the bench.
Touches of class from Jonathan de Guzman and Michu - who was at times wasteful - then saw Swansea 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 missed 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.
In the second half, 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.
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.
A deep 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.