The midfielder drilled home inside 66 seconds and added a second with a calm finish 104 seconds later - the fastest a player has ever netted a brace in the competition.
Bruno Fernandes and Victor Lindelof joined the scoring at Old Trafford, before Leeds defender Liam Cooper pulled one back prior to half-time.
Daniel James and Fernandes - this time from the penalty spot - then made it six after the break, the first time Man United had tallied a half-dozen in the league since the August 2011 8-2 demolition of Arsenal.
Ole Gunnar Solskjaer's side climbed to third with the emphatic win, although it was Leeds' Stuart Dallas who rounded off the scoring with a consolation, bringing an end to a barmy encounter.
Man United was quick out of the blocks in the first league meeting between the two rivals in over 16 years, McTominay firing a low first-time shot past Illan Meslier after Fernandes carried the ball forward and teed up his team-mate.
Anthony Martial held off Luke Ayling and played in McTominay for a quickfire second, the Scotland international picking out the same bottom-right corner with his precise finish.
Leeds created some promising opportunities and Patrick Bamford correctly had a goal chalked off for offside, but they were three down when Fernandes ran onto a loose ball and fired across Meslier.
The visitor have made a habit of conceding from set-pieces this season and was again undone by a simple delivery into the box, with Martial helping on Luke Shaw's corner for Lindelof to poke in.
Cooper gave Leeds just a glimmer of hope with a header that David De Gea, restored to the line-up in place of Dean Henderson, could not keep out despite getting a hand to the ball.
Former Leeds transfer target James scored Man United's fifth midway through the second half, though, with a shot through the legs of Meslier after taking McTominay's pass - intended for Fernandes - in his stride.
Fernandes then piled on the misery from the penalty spot following substitute Pascal Struijk's challenge on Martial inside the box, rendering Dallas' sensational goal from 25 yards nothing more than a consolation.
Meanwhile, Sam Allardyce's return to the Premier League as West Brom manager got off to a terrible start as the Baggies were outclassed 3-0 by local rival Aston Villa, which moved up to ninth.
Anwar El Ghazi put Villa ahead inside five minutes before Jake Livermore's red card for a lunge on Jack Grealish gave Allardyce's men a mountain to climb.
Dean Smith's side dominated the entire match but had to wait until six minutes from time to double their lead when Grealish teed up Bertrand Traore for a composed finish before El Ghazi rounded off the scoring from the penalty spot after Grealish had been fouled in the box.
West Brom remains second bottom, three points off safety.
Bottom-of-the-table Sheffield United doubled its points tally for the season but was cruelly denied a first win of the campaign after Brighton's Danny Welbeck cancelled out Jayden Bogle's opener in a 1-1 draw.