Cumbrians striker Danny Graham almost profited from hesitancy by Forest centre half Wes Morgan early on in Tuesday’s League One clash at Brunton Park but he screwed his shot wide.
Graham tested Paul Smith with a raking 30-yard shot which the Reds goalkeeper parried to safety.
McCleary, a half-time substitute for Nathan Tyson, set up Lewis McGugan for a shot which zipped just wide.
Forest broke the deadlock after 75 minutes when McGugan curled in a free kick and Chambers burst in behind the Carlisle defence to power home a header from six yards.
McCleary grabbed a second for the visitors in added time firing home from six yards after fellow substitute Emile Sinclair’s shot had been blocked.