City's chances of reeling in Liverpool are being hampered by injuries in defence and the loss of Kevin De Bruyne, who is injured again after a run of problems last season.
Manchester United manager Ole Gunnar Solskjaer is under pressure after a third league defeat left his side two points above the relegation zone.
Nineteen-year-old Longstaff, who clipped the crossbar in the first half, sealed a dream debut with the winner in the 72nd minute. Jetro Willems ran at the static United defence and cut the ball back for Longstaff, who let fly from outside the penalty area.
The visitors were without the likes of Paul Pogba, Anthony Martial and Jesse Lingard.
"We do lack quite a few key players but that's no excuse now," Solskjaer said. "The players work hard, they ran their socks off. At the moment, we're in a position we're not used to."
Counterpart Steve Bruce enjoyed his first home victory as Newcastle manager — and his first win as a coach over his former side United at the 23rd attempt across all competitions.
Chelsea beat Southampton 4-1 away to stretch the team's winning run to four games across all competitions under Frank Lampard.
Tammy Abraham, Mason Mount and N'Golo Kante scored in the first half, before Christian Pulisic brilliantly set up fellow substitute Michy Batshuayi for Chelsea's fourth goal in the 89th minute.
Former Chelsea defender David Luiz helped Arsenal climb to third with his first goal for the Gunners in a 1-0 win over Bournemouth, a header from a ninth-minute corner.
- AP