Maidstone manager George Elokobi said the "magic of the FA Cup is very much alive" after the sixth-tier side stunned Championship high-flyers Ipswich 2-1, while Luton struck at the death to dump Everton out in the fourth round on Saturday.
Vitaliy Mykolenko's own goal gave Luton a half-time lead that was cancelled out when stand-in goalkeeper Tim Krul allowed Jack Harrison's shot through his grasp early in the second period.
Joao Pedro scored a hat-trick, including two penalties, as Brighton beat Sheffield United 5-2 at Bramall Lane.
Newcastle, who last won the FA Cup 69 years ago, eased to a 2-0 victory at Fulham thanks to goals from Sean Longstaff and Dan Burn.
Championship leaders Leicester made light work of Birmingham as Jamie Vardy was on target in a 3-0 victory.
Leeds and Plymouth is one of at least four ties set for a replay after they drew 1-1 at Elland Road.