It gave Billy Davies' side a 10th successive home league win and moved them back into second place in the table after rivals West Brom were beaten at QPR.
Swansea remain fourth but will feel they were robbed of the chance to narrow the gap on Forest after being denied what seemed to be a stonewall penalty.
Former Forest target Darren Pratley had his heels clipped by Chris Cohen but referee Colin Webster waved away the appeals.
George Boyd was handed his Forest debut as one of three changes made by Davies.
The loan signing from Peterborough came into the side as a winger as part of a 4-5-1 formation in place of joint-leading scorer Robert Earnshaw.
Guy Moussi returned in the centre of midfield to fill in for Cohen, who reverted to full-back at the expense of James Perch. Chambers replaced Kelvin Wilson at the heart of the defence.
Swansea made two changes. One was enforced following Ferrie Bodde's season-ending knee injury against Peterborough last weekend. David Cotterill was also absent as Thomas Butler and Mark Gower came into the side.
Boyd almost made a dream start to his City Ground stay but struck his shot into the side-netting early on.
Forest had the ball in the net soon afterwards but Dexter Blackstock's effort was ruled out for offside.
Davies' side suffered a setback after seven minutes when skipper Paul McKenna was forced off with what looked to be a knee injury. Lewis McGugan replaced him in midfield.
Both keepers were forced into action for the first time in quick succession midway through the first half.
Lee Camp produced a fine save to thwart Shefki Kuqi before Dorus De Vries denied Boyd a debut goal with an equally good save at the other end.
Cohen should have put Forest in front before the break when he ghosted in unmarked at the far post but he somehow volleyed Paul Anderson's cross over from six yards out.
Pratley just failed to connect with a cross when he was alone in space early in a second half where chances were few and far between.
Forest escaped a penalty shout, and Cohen a possible red card, as it looked like the game was destined to finish goalless.
However, Chambers sent the City Ground into raptures in the first of four minutes of injury time with a timely first league goal of the season.