An early Richarlison goal and Gylfi Sigurdsson's second-half penalty ended a wretched run for the Toffees at Liverpool's home, winning at Anfield for the first time since September 1999.
The result also brings up an embarrassing stat for the champions, who have lost four league home matches for the first time since 1923.
It puts Everton level on points with Liverpool and Ancelotti's men have played a match less.
Sloppy Liverpool created many chances but were thwarted by the likes of Jordan Pickford and Michael Keane.
They'll also be sweating on the fitness of Jordan Henderson, who hobbled off in the first half holding both his hamstrings and groin on a night where just about everything that could have gone wrong for Liverpool did.
But full marks must go to Everton, who set up perfectly with talkSPORT's Sam Matterface labelling it a 'tactical victory' for Ancelotti.
Ancelotti will be full of praise for his players though, particularly Pickford, who made a number of great saves, including a Henderson volley in the first half and a close range effort to deny Mohamed Salah in the second.
Pickford's opposite number Alisson kept out a point-blank header from Seamus Coleman but could do nothing about Richarlison's opener.
The Brazilian got goalside of Ozan Kabak as he latched onto James Rodriguez's incisive pass before confidently drilling the ball home.
Liverpool upped the ante after the break but couldn't break through. Keane and Mason Holgate made brilliant last-ditch clearances to deny the lively Sadio Mane.
Everton then got their second as Sigurdsson stroked the ball in from 12 yards out. The awarding of the spot-kick was controversial with Alexander-Arnold penalised for bringing down Dominic Calvert-Lewin, despite not knowing much about it.
Referee Chris Kavanagh was advised to look at the incident again on his pitchside monitor but barely glanced at the screen as he stuck to his guns, something that didn't sit well with Steve McMahon.
He said on commentary for talkSPORT: "He's taken no notice at the monitor at all, but that's what it's there for. VAR is failing in so many ways!"
Carlo Ancelotti was hailed for a 'tactical victory' over Jurgen Klopp as Everton claimed a monumental 2-0 win over Liverpool at Anfield.