Jesse Lingard scored his fourth goal in as many games and substitute Romelu Lukaku added a stoppage time effort as Manchester United reached the English FA Cup fourth round.
The former champions beat Championship side Derby County 2-0 on Friday.
Lingard, who scored a pair of fine goals in a 2-2 Premier League draw with Burnley followed by another in a 2-0 win at Everton, produced another exquisite finish this time.
He unleashed a first-time shot from 18 metres in the 83rd minute.
Lingard hit the post several minutes earlier as Derby looked set to earn a reply after United had missed a string of chances.
Marcus Rashford also hit the woodwork while visiting goalkeeper Scott Carson produced a barrage of good saves.
Lukaku, who returned from a head injury and came on as a halftime substitute for Henrikh Mkhitaryan, broke away and powered his shot past Carson at the death.
This came as Derby threw men forward in search of an equaliser.
In a related development, Virgil van Dijk made a dream start to his Anfield career on Friday, scoring the winning goal on his debut for Liverpool.
They beat city rivals Everton 2-1 in their 2017/2018 English FA Cup third round tie at Anfield.
The Dutchman, signed last week from Southampton for a reported fee of 75 million pounds, rose inside the area to head home an Alex Oxlade-Chamberlain corner kick.
The 84th minute goal provided a relief in what had been a pulsating Merseyside derby.
Liverpool looked to have been frustrated after Gylfi Sigurdsson brought Everton level in the 67th minute after a brilliant counter-attack from Sam Allardyce’s side.
The home side had taken the lead through a 35th minute penalty kick from James Milner after Everton defender Mason Holgate had pulled down Adam Lallana in the box.