League Cup Final: Liverpool scored all 11 of their penalties but Chelsea goalkeeper Kepa Arrizabalaga, who came on as substitute for the shootout, failed to convert his spot-kick.
James Milner was "gutted" after Liverpool "didn't get the job done" in their 3-2 FA Cup loss to Manchester United on Sunday.
Roberto Firmino scored twice from Trent Alexander-Arnold crosses either side of James Milner's penalty before the brilliant Alexander-Arnold rounded off the scoring himself.
James Milner said he would be delighted if Manchester United did Liverpool a favour in the Manchester derby at Old Trafford.
Liverpool now hold a two-point lead over City, but the defending champions have a game in hand.
Despite late drama, victory was well deserved for Liverpool who dominated throughout and bounced back from blowing a 2-0 first-half lead.
James Milner, Adam Lallana and Nathaniel Clyne took it upon themselves to inspire Alastair Cook & Co. ahead of the third Ashes Test.
England midfielder James Milner, who joined on a free transfer last month from Manchester City, was on target in the 67th minute for the second time in four days for Liverpool in Australia.
James Milner, the 29-year-old, capped 53 times by his country, will join Liverpool on July 1 after his Manchester City contract expires. Liverpool have not disclosed the length of his contract.
As Manchester City look for a second title in three seasons, Milner feels that the visit of Liverpool, and the following home game against Crystal Palace, offer City a real opportunity to take charge in the title race.