In a match that Liverpool went on to dominate after a lacklustre first-half display, Richarlison's third-minute goal and Gylfi Sigurdsson's late penalty proved enough for Everton to claim a first away win over their Merseyside rivals since September 1999.
Liverpool's efforts were not helped by the loss of Jordan Henderson to injury, with Sadio Mane missing two presentable chances that might have made Everton pay for profligate finishing from Seamus Coleman and Richarlison.
The centre of derby drama in recent meetings, Jordan Pickford also played a pivotal role, making a wonderful save from Mohamed Salah, setting the stage for Sigurdsson to condemn Jurgen Klopp's champions to a fourth successive home reverse.
Everton made a blistering start in the blustery conditions – James Rodriguez slipping a pass through to Richarlison, who finished crisply across Alisson and into the left corner to put the Toffees ahead in a Merseyside derby for the first time since 2010.
Henderson had a winner disallowed by VAR in October's reverse fixture, and his wicked volley would have restored parity if not for Pickford's finger-tip save, with Everton's goalkeeper then denying Trent Alexander-Arnold.
Liverpool were dealt a blow on the half-hour as Henderson succumbed to injury, and Coleman should have compounded the hosts' frustration when he met Lucas Digne's cross, only to head straight at Alisson.
Mane twice missed from close range early in the second half, before Michael Keane's last-ditch tackle denied Liverpool's number 10.
A rare Everton attack resulted in Richarlison racing behind Liverpool's defence, only for the Brazilian to hesitate when one-on-one with Alisson.
Salah seemed set to punish Everton when Liverpool carved out another chance, but Pickford rushed out to smother the angle.
And that proved vital, with Calvert-Lewin soon adjudged to have been fouled by Alexander-Arnold at the other end of the pitch after Alisson's fine save.
After a long wait for a VAR check, Sigurdsson slotted in the resulting penalty as Carlo Ancelotti masterminded the Toffees' first derby victory in over a decade.