The Norwegian was assisted by Kevin De Bruyne three times in the first 40 minutes at Kenilworth Road to complete the 20th hat-trick of his club and country career, including his eighth triple since joining the Citizens - this one being his first away from the Etihad Stadium.
A close-range finish in the fourth minutes was followed by two almost identical bursting runs through on goal with two completely different finishes, the first a powerful strike under the legs of Tim Krul before delicately dinking the ball over the Hatters goalkeeper to add a third.
Jordan Clark gave Luton a glimmer of hope with two superb strikes either side of half time to reduce the deficit to one, but after meeting a wonderful long-range through-ball from Kyle Walker, De Bruyne once again teed up Haaland for net City's fourth of the night from a few yards out on the 55-minute mark.
Three minutes later, Haaland scored his fifth, receiving a pass from Bernardo Silva this time before unleashing a fierce shot that powered straight through the gloves of Krul.
Mateo Kovacic then got in on the act to with a swerving 20-yard strike to rub salt into Luton's wounds and help FA Cup holder Man City secure safe passage through to the quarter-finals.
Not only has Haaland matched his five-goal haul from a Champions League tie against RB Leipzig last season, the 23-year-old has also become the first player for a top-flight club to score at least five goals in an FA Cup game since former Manchester United man George Best netted six against Northampton Town back in 1970.
City's No.9 has also equalled an FA Cup club record as his five-star goal tally has put him alongside Frank Roberts and Bobby Marshall for the most goals netted by one player in a single match in the competition.
Haaland has now scored a remarkable 79 goals in 83 games since joining Pep Guardiola's side in the summer of 2022, including 27 strikes in 30 appearances across all competitions so far this season.
Meanwhile, Haaland's partner in crime De Bruyne has registered 12 assists in all tournaments this season, despite spending around six months on the sidelines through injury.
Discussing his connection with De Bruyne after the match, Haaland told ITV: "Of course, [he has] massive [influence]. Look at what he's doing! He's doing what he's best at. It's a pleasure to play with him.
"It's quite good [connection]. I think we know what we both want from each other. We look at each other often throughout the game. It clicks well. He's a smart guy and I like to play with smart guys.
Haaland added: "My fitness is getting there and I'm getting back to my best finally. I'm feeling good and it's an amazing feeling.
"I think (as a team) we made a plan and everyone stuck to it perfectly. We did well and kept the ball. That's what we need to do against these teams. So well played by everyone.
"Yeah, [our peak is] coming. We're coming. Exciting times ahead. We are ready to attack."
Man City, who have extended their unbeaten run in all competitions to 18 games, will now shift their attention to Sunday's derby showdown at home against rivals Manchester United in the Premier League.