Goals from Ashley Barnes and Chris Wood give Burnley their first win since December in a second-half comeback for Sean Dyche's side.
Allardyce made a number of changes to his starting line-up, including handing a first Premier League start in weeks to January signing Cenk Tosun.
Despite a tense affair early on, Tosun opened the scoring after 20 minutes with a trademark Everton striker’s header.
The Blues had a free kick just inside the Burnley half but that was easily cleared by the Clarets backline. Instead, Gylfi Sigurdsson recycled the loose ball out wide to Theo Walcott and the England international was allowed to cross.
His ball was flicked on by Seamus Coleman over two Burnley defenders and into the path of Tosun who nailed his header low past Nick Pope.
The hosts drew level 10 minutes into the second-half.
Barnes, who troubled the Blues all game, latched onto a ball through the right-hand channel and beat Michael Keane for pace. He bore down on Jordan Pickford and rifled his effort into the top of the Everton net.
Wood, the half-time substitute, battered home a header from a corner with 10 minutes to go to give Dyche's side all three points.
Visitors have early going
Despite naming two strikers in the starting line-up, Allardyce opted for Dominic Calvert-Lewin to play on the left whilst Sigurdsson sat behind Cenk Tosun.
Having the Icelandic international playing in his favoured role seemed to work early for Allardyce's side as they created a handful of early goalbound efforts.
A lack of communication denied Everton taking an early lead. Sigurdsson wriggled into the Burnley box with a number of clever touches and played a ball across the area. Walcott skied the effort as he and Tom Davies ran into each other.
Davies, who was running onto the ball, had the better angle to drill the ball at Nick Pope but Walcott, angling onto the pass from a wide position, took the effort and didn’t make good enough contact.
Blues troubled by direct play
The Blues struggled with the Clarets direct play in the first 45. When Dyche’s side went direct towards Barnes, Keane and Ashley Williams struggled to cope. When Matthew Lowton and Aaron Lennon crossed from the right, Keane and Williams struggled to cope.
It was that type of play from Lowton that almost the hosts level.
An outrageous save from Pickford kept the score in the favour of the visitors. Ashley Barnes’ goal-bound header meant the ‘keeper had to be at full stretch to keep the ball out.
He was and he did.
Hosts try and exploit the weaknesses
The introduction of Chris Wood at half-time showed that Dyche had spotted the glaring weakness of the Everton defence.
Burnley’s record signing held the ball up well and drew the attention of Keane and Williams, allowing Barnes to operate in the space - regularly attacking Cuco Martina on the left-hand side.
It was that style that allowed Barnes to draw Dyche's side level early into the second period.
The Clarets nearly took the lead through Wood but his goal-bound header from a corner just before the hour mark was smartly saved by Pickford.
Wood would get another chance from a corner with 10 minutes to go and this time Pickford stood no chance of keeping his header out.
As the Blues attempted to draw level, Williams was given his marching orders by referee Chris Kavanagh for a stray elbow in the Burnley penalty box.
The result sees Everton remain winless away from Goodison Park since December as Allardyce continues failing his bid to turnaround the club's away day blues.
As he said in his pre-match press conference, the Blues only won two games away during the calendar year of 2017 and have yet to win away in the new year.
Dyche's side will move on to 40 points, more or less securing another season in the Premier League.