Chambers' intervention secured their tenth consecutive win at the City Ground and aided their Premier League promotion aspirations.
The Welsh side, who went into the game in fourth place, thought they had claimed a point after a string of fine saves from goalkeeper Dorus de Vries kept the home side at bay.
But it was Chambers who rose at the far post in the 91st minute to force home a Lewis McGugan corner and kick-start wild celebrations.
New signing George Boyd almost had an instant impact following his move from league rivals Peterborough United.
In the opening minute, his mis-hit cross almost caught de Vries out but the stopper recovered to save the day.
Darren Pratley, the Swansea midfielder Forest had tried to sign in January, came close with a low shot that flashed across goal.
Dexter Blackstock had a goal chalked off for offside, after Boyd had sent him clear during a frenetic start to the contest.
Forest suffered an early blow as skipper Paul McKenna limped off with an injury just seven minutes in, to be replaced by Lewis McGugan.
A Blackstock flick gave Boyd the chance to try his luck just inside the box, but his shot was high and wide.
Forest were furious with referee Colin Webster after Shefki Kuqi physically bundled Wes Morgan out of the way to break into the penalty area, with Lee Camp quick off his line to make a vital save as the match official allowed play to continue.
But it took a breathtaking save from De Vries at the other end to deny Forest the lead.
A blocked shot from Guy Moussi fell perfectly for Boyd but his driven effort was flicked away brilliantly by the keeper.
The Swansea stopper was back in action against moments later, making a double save to twice deny Blackstock from close range.
Chris Cohen was then guilty of missing an outstanding opportunity after Paul Anderson had weaved his way into space down the right.
Anderson picked out the full-back at the far post, but with just the Swans keeper to beat, he fired wide.
The second half saw both sides struggling to find the same cohesion as they had before the break, although one flowing move saw Gunter surge down the right and set up a chance for Majewski, who shot over.
Neat footwork from McGugan then saw him dance through two challenges into the box, but keeper De Vries was quick to smother the ball once again before the midfielder could get a shot away.
Swansea were denied a clear cut penalty when Pratley had his heels clipped as Chambers and Cohen pursued him into the box, with referee Webster waving play-on.
And Chambers rubbed salt into the wound in injury time as he rose to connect with a McGugan corner and break Swansea's hearts at the death.