The win has moved Jurgen Klopp's side into fifth position in the table, four points behind fourth-placed Chelsea with a game in hand, while Man United remain second, four points clear of third-placed Leicester City.
The visitors were awarded a penalty in the 26th minute when Eric Bailly clattered into Phillips, but the referee overturned his decision, having had a look at the monitor, with the Man United centre-back getting a slight touch on the ball before contact was made with the 24-year-old.
Liverpool were level in the 34th minute, though, when Phillips's low effort was turned into the back of the net by Jota, who was bright for the Reds in the opening half of the contest.
The Merseyside giants then took the lead in the closing minutes of the first half when Firmino headed a brilliant Trent Alexander-Arnold cross into the back of the net at the far post.
Firmino doubled his tally for the night early in the second period to send Liverpool 3-1 ahead, with the Brazilian converting from close range after Dean Henderson failed to deal with a strike from Alexander-Arnold.
Jota hit the post just before the hour, while Alexander-Arnold also came close during a period where the visitors were utterly dominating the contest in Manchester.
The home side could not find a third despite some late pressure, though, and Liverpool made sure of the points in the final exchanges courtesy of a cool finish from Mohamed Salah.
Man United are back in Premier League action against already-relegated Fulham on May 18, while Liverpool will be attempting to put another victory on the board when they travel to West Bromwich Albion, who are also already down, on Sunday.
3rd min: Big moment early on as Firmino breaks into the Man United box before attempting to pick out Jota with a low cross; the ball hits Bailly's hand, but the referee is not interested, and VAR agree, which is not a surprise considering what actually occurred.
5th min: Huge chance for Man United in the fifth minute as Alisson Becker kicks the ball straight to Cavani, but the striker misses the target from a dangerous position; what an opportunity for the 20-time English champions to make the breakthrough here.
10th min: Man United make the breakthrough in the 10th minute of the contest as Fernandes's strike hits Phillips and flies into the back of the net; big moment early on at Old Trafford.
23rd min: Chance for Liverpool to level the scores as Alexander-Arnold delivers a wonderful cross into the Man United box, Jota makes contact, but Dean Henderson makes a smart save.
26th min: Liverpool are awarded a penalty as Bailly clatters into Phillips; VAR have a look at the incident and decide that the referee needs to have a look at the monitor; the official heads across and overturns his decision, with Bailly getting a slight touch on the ball.
31st min: Paul Pogba loops a header just wide of the Liverpool post as Man United threaten to score a second as the first period continues to gather pace at Old Trafford.
33rd min: Smart save from Man United goalkeeper Dean Henderson to keep out a powerful strike from Jota, who has been bright for Liverpool in the opening 33 minutes.
34th min: Liverpool level the scores in the 34th minute of the contest as Jota turns an effort from Phillips into the back of the Man United net from close range.
41st min: Pogba works his way into the Liverpool box before striking towards goal, but the midfielder's powerful effort was always the wrong side of the post.
45th min: Liverpool take the lead in the final stages of the first period as Firmino heads a brilliant free kick from Alexander-Arnold into the back of the net at the far post.
47th min: The visitors score a third in the early stages of the second period as Henderson palms an Alexander-Arnold effort straight into the path of Firmino, who converts from close range.
59th min: Liverpool have another huge chance as the ball is worked into Jota inside the Man United box, but the Portugal international sees his effort come off the post.
62nd min: Henderson keeps out a low strike from Alexander-Arnold as Man United continue to struggle, with gaps appearing all across the home side's defence at Old Trafford.
68th min: Man United score a second to keep the match alive as Cavani's clever pass releases Rashford, and the England international manages to squeeze the ball into the bottom corner of the goal.
70th min: Huge chance for Man United to level as the ball bounces around inside the Liverpool box, with Mason Greenwood having two separate opportunities, but Phillips clears off the line before the Man United attacker sees his next effort blocked.
75th min: Henderson does well to prevent Mane from rounding him and surely finishing into the back of the net as Liverpool threaten to put the game to bed.
76th min: Rashford does brilliantly to move past Phillips before striking towards goal, but the England international's effort was always the wrong side of the post.
90th min: Salah makes sure of the points for Liverpool as he races clear before comfortably slotting the ball into the bottom corner of the net past Henderson; Liverpool have won this.
90th min: Rashford has the chance to score again for Man United in the final moments, but the attacker slots his low effort just wide of the far post.
MAN OF THE MATCH - ROBERTO FIRMINO
It would be fair to say that Firmino has not had his best season for Liverpool, but the Brazilian was in excellent form at Old Trafford as he played a leading role in the visitors securing all three points.
The 29-year-old scored either side of the half-time interval to turn the match on its head, but his overall work was also excellent, with Jota enjoying plenty of space throughout the game due to his teammate's work.
Anthony Taylor had initially awarded Liverpool a penalty in the 26th minute when Bailly flew into Phillips, but the referee then overturned his decision, having had a look at the monitor at the side of the pitch.
Replays showed that Bailly got a slight touch on the ball before he connected with the Liverpool centre-back inside the box, but it was still a heavy challenge from the Ivory Coast international and certainly a major talking point in the contest.
Possession: Man United 52%-48% Liverpool
Shots: Man United 17-17 Liverpool
Shots on target: Man United 3-8 Liverpool
Corners: Man United 6-4 Liverpool
Fouls: Man United 10-12 Liverpool
MAN UNITED (4-2-3-1): Henderson; Wan-Bissaka, Lindelof, Bailly (Matic 86'), Shaw; Fred (Greenwood 63'), McTominay; Rashford, Fernandes, Pogba; Cavani
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Phillips, Williams, Robertson; Fabinho, Thiago, Wijnaldum (Jones 74'); Salah (Williams 92'), Firmino, Jota (Mane 74')