In their first top-flight campaign since 1992, Luton's primary goal is survival and a draining replay was the last thing Hatters boss Rob Edwards needed.
Forest were also given a scare by League One opponents as Nuno Espirito Santo's men recovered from two goals down to draw 2-2 with Blackpool.
Blackpool beat Forest in the third round last year when the Seasiders were in the Championship and almost repeated the trick at the City Ground.
Forest academy graduate Jordan Lawrence-Gabriel netted against his old club with a spectacular diving header in the 25th minute
Albie Morgan doubled Blackpool's lead in the 27th minute, coolly converting CJ Hamilton's cross at the far post.
But Nicolas Dominguez finished off Gonzalo Montiel's cross in the 39th minute and Morgan Gibbs-White equalised from 18 yards in the 56th minute.
West Ham were held to a 1-1 draw against second-tier Bristol City at the London Stadium.
Jarrod Bowen ran onto Lucas Paqueta's lofted pass and slotted home in the fifth minute.
But influential playmaker Paqueta was forced off with an injury midway through the first half in a major blow.
And Tommy Conway sent 9,000 travelling City fans wild in the 61st minute when he fired the equaliser past Lukasz Fabianski.
Later on Sunday, Arsenal and Liverpool take a break from their Premier League title battle to clash in the tie of the round at the Emirates Stadium.
After the title rivals played out a 1-1 draw in the Premier League at Anfield in December, they meet again with the chance to inflict a psychological blow.
