A crazy game at Upton Park saw the hosts go ahead twice, the second time a penalty from Mark Noble, who was sent off shortly afterwards for a second yellow card
Young English midfielder Ravel Morrison gave the hosts the lead at Upton Park in the first half as his shot took a significant deflection but the left-back drew the visitors level in the 62nd minute with the first of his sublime set-piece goals.
A Mark Noble penalty looked to have secured a win for the home side with 14 minutes remaining but the midfielder was then dismissed after picking up a second yellow card and Baines fired in the resulting free kick before debutant Lukaku headed home the winner.
Sam Allardyce, whose side have not won since the opening day, was forced into one change from the starting XI who drew 0-0 at Southampton in their last fixture as the injured Guy Demel was replaced in defence by debutant Razvan Rat, a close-season signing from Shakhtar Donetsk.
|VIEW FROM UPTON PARK
|By Jay Jaffa
What a superb game of football that turned out to be! Sam Allardyce will wonder where it went wrong after taking the lead for the second time via a penalty but it was Lee Mason's decision at the other end of the pitch that had the biggest impact on the final score.
From my position up high and at the other end of the ground, it looked like Mark Noble got the ball. Instead he received a second yellow card and Everton were handed fresh impetus to press for an equaliser.
As it turned out, Leighton Baines almost mocked Jussi Jaaskelainen as his second goal from a dead ball whistled into the opposite corner to his first. His Belgian team-mates stole the points, though, as Romelu Lukaku - who had a superb Toffees debut - headed home Kevin Mirallas's precise cross. Great game, great comeback.
The visitors began the match confidently and after a well-weighted pass from Ross Barkley, Kevin Mirallas cut in from the left before unleashing a stinging shot that Jussi Jaaskelainen palmed away.
The hosts finally registered a shot on goal, only their seventh of the season, after 31 minutes and it handed them the lead. Matt Jarvis cut inside from the left with purpose and shaped to shoot before laying the ball off to Morrison, whose effort was taken past the helpless Tim Howard thanks to a big deflection off Phil Jagielka.
Martinez reacted at half-time by bringing James McCarthy and Chelsea loanee Lukaku on for Steven Naismith and Nikica Jelavic. He was soon forced into another change as Bryan Oviedo replaced the injured Leon Osman.
Baines found his range just past the hour as Barkley won a free kick just outside the area and the 28-year-old stepped up and curled a magnificent strike into the top corner of the goal.
Everton were not level for long, however. McCarthy tripped the rampaging Kevin Nolan in the penalty area 15 minutes from time and Noble converted from the spot.
A few minutes after being the hero, Noble found himself walking down the tunnel. A wild lunge on Barkley saw him given his marching orders for his second yellow card.
It proved costly as Baines once again stepped up and fired the free kick home after 83 minutes, this time opting for the other corner and placing it home off the post.
And just two minutes later, Mirallas broke free down the right and crossed for Lukaku, who powered a header past Jaaskelainen to take Everton up to fourth.