City lost 3-2 at Aston Villa last weekend in a surprise result but managed to recoil positively by coming from behind to beat the previously undefeated Toffees at home.
Everton's on-loan Chelsea striker Romelu Lukaku scored his fourth goal in three appearances when he skipped around his marker before slotting home underneath Joe Hart in the 16th minute before Alvaro Negredo responded with a scuffed effort from Yaya Toure's through ball a minute later.
The American goalkeeper was desperately unlucky to concede a third when he had saved Aguero's 69th minute penalty, after Pablo Zabaleta had gone down easily under pressure from Seamus Coleman. The ball bounced off the upright and back off his head to hand City all three points to send them fifth on the table.
Luis Suarez made it three goals in two Premier League outings since returning from a 10-match ban when he slotted home despite falling to his knees from just outside the six-yard box in the 13th minute.
The Reds soon doubled their lead when Daniel Sturridge jumped one goal clear at the top of the scoring chart with his sixth goal of the term, made possible by a weaving effort and classy angled finish.
A third came late in the first half when Raheem Sterling was controversially awarded a penalty despite minimal contact from Dean Moxey allowing skipper Steven Gerrard to step up and send Julian Speroni the wrong way.
Palace came up with a consolation goal from Dwight Gayle in the 76th minute but the damage had already been done as Brendan Rodgers' side stayed second on the table, behind Arsenal by virtue of goals scored.
Elsewhere, Loic Remy scored twice as Newcastle United held off Cardiff City 2-1 in Wales, Manchester United arrested a two-game losing streak by coming from behind to beat Sunderland 2-1 thanks to a quickfire brace from Adnan Januzaj, Fulham edged Stoke City 1-0 at home and Hull City and Aston Villa played out a 0-0 draw.
Sunday's fixtures saw Chelsea produced a late show to beat Norwich City 3-1 at Carrow Road, West Ham United shocked Tottenham 3-0 at White Hart Lane in the London derby, Southampton moved to fourth with a 2-0 home triumph over Swansea City and West Bromwich Albion held league leaders Arsenal to a 1-1 draw at The Hawthorns.
2013/14 Barclays Premier League scores
For breaking stories and all the great banter like us on Facebook: facebook.com/tribalfootball