Chris Wood's double had left Forest two goals ahead in the 76th minute, but Chiedozie Ogbene halved the deficit in the 83rd minute before Elijah Adebayo scored a leveller in the second minute of added time at the end of the 90 minutes.
Luton are now 17th in the Premier League table on five points, while Forest sit 15th, boasting 10 points from their opening nine matches of the campaign.
Wood had the first half-chance of the match in the sixth minute, but the Forest striker's header was always comfortable for Thomas Kaminski between the Luton sticks.
The home side came close again in the 23rd minute, with the ball falling kindly for Harry Toffolo inside the penalty box, but Kaminski managed to make another smart save.
The clearest opening of the first period also came for Forest in the closing stages of the half, but Wood could only head over the crossbar from close range.
Ibrahim Sangare then fired wide of the target in the last piece of action in the opening 45 minutes after Kaminski had palmed a cross from Morgan Gibbs-White into a dangerous position.
Forest dominated the first half in terms of the possession, having 66% of the ball, while they had eight attempts, but Luton were able to enter the interval all square.
The home side broke Luton's resistance early in the second period, though, with Gibbs-White and Anthony Elanga combining before the ball came for Wood, who found the back of the net with a clever finish.
Wood had another chance shortly after, but the forward lifted his header over the crossbar from inside the box, before Jacob Brown passed up an excellent opportunity from a header down the other end, placing his effort straight at Matt Turner.
Gibbs-White then fired over the Luton crossbar just past the hour after smart work from Sangare, with Forest threatening to double their advantage.
Wood was the standout player on the afternoon, and he scored his second in the 76th minute, with the forward planting a brilliant header into the back of the net after meeting a cross from Elanga.
Luton had one back in the 83rd minute, with Ogbene firing the ball into the top corner from inside the box after Forest had failed to deal with a free kick into their box.
The visitors then incredibly levelled the scores in stoppage time, with Adebayo using his chest to control a high pass inside the box before finding the back of the net to spark jubilant scenes in the Luton dug-out.
Wood thought that he had won it in the fifth minute of added time at the end of the 90, but the forward's goal was disallowed for offside in an incredible finish to the contest.
Forest will continue their league campaign against Liverpool at Anfield next Sunday, while Luton will be in action on the same afternoon away to Aston Villa.