United score five to start life after Ten Hag in style, Spurs dump City out of EFL Cup
Arsenal and Liverpool were among the other sides to book their place in the last eight, but Chelsea were beaten 2-0 at Newcastle.
A United legend as a player, Van Nistelrooy was thrust into the role of interim manager after the Red Devils sacked Erik ten Hag on Monday following a dreadful start to the season.
Sporting boss Ruben Amorim is the man the English giants have targeted to take over but are still in negotiations with the Portuguese champions to agree compensation for the 39-year-old.
According to reports in Portugal, Amorim will remain at Sporting for the next three matches before completing his move during the November international break.
Van Nistelrooy said before the game that United "can be unstoppable" when "players, staff and supporters pull together" and they cut loose by scoring four times in a thrilling first 45 minutes.
Casemiro's stunning effort into the top corner opened the scoring before Alejandro Garnacho swept home Diogo Dalot's cross to please a jubilant Van Nistelrooy on the touchline.
Bilal El Khannoussi quickly pulled a goal back for the much-changed Foxes.
Bruno Fernandes' deflected free-kick restored United's two-goal cushion before Casemiro slammed home his third goal in two games.
Conor Coady grabbed another consolation for Leicester but United were not to be denied just a second win in nine games.
Fernandes rounded off the scoring when he pounced on a short back-pass to round Danny Ward and fire home.
City paid for leaving Erling Haaland out in north London as Tottenham moved a step closer to ending their 16-year trophy drought.
Timo Werner slotted home his first goal since March from Dejan Kulusevski's inviting cross.
Pape Sarr's long-range strike doubled Spurs' lead before Matheus Nunes got City back into the tie in first-half stoppage time.
But they could not find an equaliser and Pep Guardiola showed where his priorities lie as Haaland remained on the bench for the full 90 minutes.
Gakpo leads Liverpool
Cody Gakpo was the match-winner for Liverpool in a 3-2 win at Brighton to continue Arne Slot's excellent start to life with the Reds.
Gakpo has not scored in the Premier League or Champions League this season but now has four goals in two League Cup games.
The Dutch international was ruthless with two powerful finishes cutting inside onto his right foot from the left wing.
Luis Diaz was also on target for Liverpool, while Simon Adingra and Tariq Lamptey struck late on for Brighton.
Teenager Ethan Nwaneri caught the eye with a sensational strike as Arsenal eased to a 3-0 win at Championship side Preston.
Gabriel Jesus and Kai Havertz got the other goals for the Gunners.
Newcastle gained revenge for their Premier League defeat at Chelsea at the weekend to ease the pressure on manager Eddie Howe.
The Saudi-backed Magpies are winless in five in the league but are desperate to end a 55-year wait to win a major trophy.
Alexander Isak's strike and an Axel Disasi own goal inside three first-half minutes proved decisive at St. James' Park.
Crystal Palace backed up their first Premier League win of the season on Sunday by beating Aston Villa 2-1 thanks to goals from Eberechi Eze and Daichi Kamada.