Chris Hughton's third placed side moved to within one point of second placed Middlesbrough and two of Championship leaders Burnley after a dramatic finale at the City Ground.
Lewis Dunk's 27th minute header had given Brighton the lead, only for Forest to equalise through Dexter Blackstock's header five minutes into the second half.
Former Chelsea midfielder Sidwell came off the bench with six minutes remaining and made an immediate impact when he fired low into the far corner from 10 yards after Forest failed to stop Anthony Knockaert advancing into the penalty area in the first minute of stoppage-time.
Middlesbrough can reestablish a four-point advantage over Brighton if they win their game in hand at home to Reading on Tuesday.