In the day's other match, Ronald Koeman's Southampton climbed to seventh in the table after first-half goals from Steven Davis and Graziano Pelle secured a 2-0 home win over south-coast rivals Bournemouth.
Everton were 2-0 up inside 31 minutes against Sunderland thanks to Kone and Spanish forward Gerard Deulofeu.
Kone sent a looping ball over the defence for Deulofeu to score the opener and the Ivory Coast international added a second goal following a one-two with strike partner Romelu Lukaku on the edge of the box.
Jermain Defoe inspired a Sunderland rally, volleying their first goal in stoppage time before playing a part in Steven Fletcher's headed equaliser five minutes after the break.
The visitors, however, were always vulnerable to the counter attack and Everton made it 3-2 when Sebastian Coates prodded a right-wing cross from Deulofeu into his own net.
Deulofeu also set up the fourth for Lukaku, with the Belgian rounding goalkeeper Costel Pantilimon to score.
Another Everton counter-attack led to Kone planting the ball into the net with his left foot in the 62nd minute before the Ivorian completed the rout with a close-range header following a centre with the outside of his left boot from Lukaku.
Everton's victory lifted them to ninth with 16 points from 11 games while Sunderland remained second from bottom on six points. Southampton have 17 points, moving above Liverpool on goal difference.
“I thought our attacking play was exceptional,” said Everton manager Roberto Martinez. “More than anything, the understanding of how to break down a defensive system.
“Sunderland came here with a clear idea of being very difficult to break down and then try to hit us on the counter attack and I thought the way we coped with that was very impressive.”
Martinez's opposite number Allardyce disagreed, blaming his own team for their second-half capitulation.
“If Everton had scored a wonder goal you can say 'there's not a lot we can do about that', but the last four goals were all stoppable by us and there was nothing brilliant about Everton's play,” said Allardyce.
“We became so open and so easy to play through. We seemed to get carried away by the euphoria of getting the equaliser.”
Southampton went ahead against fourth from bottom Bournemouth when a sweeping passing move ended with Ryan Bertrand crossing for Steven Davis to volley home from close range in the 31st minute.
Pelle then made sure of the win with a towering header following a left-wing centre by Dusan Tadic.
The only negative for Southampton was the late sending-off of Victor Wanyama for a second bookable offence. – Reuters
Original source: Kone stars as Everton thrash Sunderland