Harry Cornick's fine finish after 23 minutes put the visitors ahead at the City Ground, capping a strong start to the match from the league's bottom club.
Joe Lolley made it 1-1 before the break, though, his effort sneaking through the grasp of Simon Sluga, who did at least save brilliantly from Joe Worrall just prior to the interval.
Sluga was powerless to keep out Lolley's swerving second, however, and Lewis Grabban's 90th-minute penalty made certain of a deserved win.
Forest is now just five points behind second-place Leeds United, with a game in hand, giving Sabri Lamouchi's men real hope of a promotion push.
Luton stays bottom, four points adrift of a Stoke City side sitting just above the relegation zone.