Jankewitz, who was handed his full debut days after asking for a transfer request to force manager Ralph Hasenhuttl’s hand for more playing time, was given his marching orders for a reckless challenge on Scott McTominay after just 80 seconds.
Southampton would succumb to an Aaron Wan-Bissaka goal on 18 minutes before Marcus Rashford doubled the lead seven minutes later.
Jan Bednarek added to the misery with an own goal on 34 minutes and Edinson Cavani made it 4-0 approaching half-time and Anthony Martial and McTominay each added goals after the break.
Bednarek was also shown a straight red card for a foul on Martial in the 86th minute, which resulted in a long Virtual Assistant Referee check and Bruno Fernandes converting the resulting penalty kick.
Saints are without a league win since their 1-0 triumph over Liverpool back on January 4, having dropped all four matches since.
Hasenhuttl was without Oriol Romeu and Ibrahima Diallo, who both limped off in the defeat to Aston Villa last Saturday, as well as Jan Valery, Jake Vokins and Shane Long — all sent out on loans.
Manchester United (4-2-3-1): de Gea; Shaw (van de Beek 46), Maguire, Lindelof, Wan-Bissaka; Fred, McTominay; Greenwood, Fernandes, Rashford (James 60); Cavani (Martial 46)
Unused subs: Henderson (GK), Matic, Pogba, Telles, Bailly, Williams
Goals: Wan-Bissaka (18), Rashford (25), Bednarek (34 OG), Cavani (39), McTominay (71), Fernandes (87 PEN), Martial (68, 90), James (90+3)
Bookings: Maguire (52), van de Beek (53)
Southampton (4-4-2): McCarthy; Bertrand, Stephens, Bednarek, Ramsay; Armstrong, Jankewitz, Ward-Prowse, Djenepo (Tchaptchet 78); Ings (Redmond 71), Adams
Unused subs: Forster (GK), Lewis (GK), N’Lundulu, Ferry, Watts, Chauke, Finnigan
Bookings: Jankewitz (2 RED), Ramsay (42), Stephens (45), Bednarek (86 RED), Armstrong (89)