Liverpool cruised to their ninth successive Premiership victory as struggling Everton's problems piled up.
Peter Crouch's clever finish and Steven Gerrard's deflected 25-yard drive put Liverpool two up inside 17 minutes.
James Beattie, who had a header controversially ruled out, pulled one back four minutes before half-time.
But Djibril Cisse fired in Liverpool's third after 47 minutes and Everton were reduced to nine men when Phil Neville and Mikel Arteta were sent off.
Beattie's header may have ended Liverpool's sequence of 761 minutes of Premiership action without conceding a goal, but it was another impressive performance from Rafael Benitez's rapidly-improving side.
Liverpool recalled Cisse and left Fernando Morientes on the bench, while Everton recalled experienced duo David Weir and Nuno Valente in defence.
And it was Liverpool who flew out of the blocks and stunned Everton with two early goals.
Cisse had already fired wide when Gerrard's header released Crouch after 10 minutes, and he expertly rounded Nigel Martyn to roll a composed finish into an empty net.
And seven minutes later, a rampant Liverpool doubled their advantage as the 202nd Merseyside derby threatened to turn into a nightmare for Everton.
Goalkeeper Martyn punched clear, but when Cahill's header fell to Gerrard 25 yards out, he fired a low shot into the corner with the help of a slight deflection off Joseph Yobo.
Cisse then shot inches over as Liverpool threatened to run riot, but Everton slowly worked their way back into the game.
They were aggrieved when Beattie had a header ruled out from Tim Cahill's cross, with replays throwing doubt on whether the ball was out of play.
But the striker was rewarded with a goal that gave Everton a lifeline four minutes before the interval.
Kevin Kilbane's far post header was cleverly turned on by Simon Davies, and Beattie headed past Pepe Reina.
But any hope Everton had of mounting a revival appeared to be snuffed out when Liverpool added a third only two minutes after the re-start.
Liverpool broke quickly, and Cisse raced on to Harry Kewell's pass before easily avoiding a poor challenge from David Weir to beat Martyn.
Everton's confidence was visibly knocked, and Gerrard could have added a fourth for Liverpool with a powerful drive that was only inches off target in the 67th minute.
And Everton's miserable night got worse a minute later when Neville was shown the red card for a second bookable offence, a foul on Mohamed Sissoko.
Beattie wasted a glorious chance to give Everton some hope in the dying minutes, but he blazed over from point-blank range from Yobo's flick.
As if Everton had not suffered enough, Arteta then also saw red for his second booking, a senseless foul on Liverpool substitute Luis Garcia.
Everton manager David Moyes:
"I felt the ball hadn't gone over the line before James Beattie's disallowed goal, but that wasn't the reason we lost.
"We conceded goals at bad times, but I'm really pleased with the effort of the boys. They gave everything they could in this game.
"The two lads who were sent off, Phil Neville and Mikel Arteta, should have known better because they knew they were on yellow cards and had to be careful.
"It's a big job and we are not doing so well, but I can't keep whipping the players. I will be there on the touchline shouting them on we will keep doing the things we think are right."
Liverpool manager Rafael Benitez:
"We knew Everton would be nervous and we said we needed to ready if they made a mistake - we did the right things in the first half and it all worked well for us.
"Djibril Cisse's goal just after half-time was good for us. We said during half-time we needed to concentrate because they might push, go too high and leave space for Cisse.
"I would like Cisse to stay here. We now have different strikers with different qualities. The question now is to use them in the right games. Cisse is a player who can give us a lot of things, as we have seen here."
Everton: Martyn, Hibbert, Yobo, Weir, Nuno Valente, Arteta, Neville, Davies (McFadden 57), Kilbane, Cahill, Beattie.
Subs Not Used: Wright, Bent, Ferrari, Osman.
Sent Off: Neville (68), Arteta (90).
Booked: Cahill, Neville, Beattie, Arteta.
Goals: Beattie 42.
Liverpool: Reina, Finnan, Carragher, Hyypia, Warnock, Gerrard (Luis Garcia 85), Sissoko, Alonso, Kewell (Riise 80), Crouch (Morientes 77), Cisse.
Subs Not Used: Dudek, Josemi.
Booked: Kewell, Crouch.
Goals: Crouch 11, Gerrard 18, Cisse 47.
Ref: G Poll (Hertfordshire).