Despite the home side dominating for almost 70 minutes, it was Middlesbrough who started the brighter of the two and opened the scoring after only eight minutes with a swift counter-attack from England international Stewart Downing.
Using his pace, the winger carried the ball from deep inside his own half straight to the heart of the Everton defence, but a good lunging tackle from Phil Jagielka rebounded straight back to the left footer.
With time to pick his pass, Downing fed an unmarked O'Neil inside the box, who side-footed the ball off the inside of the post to give his side an early lead.
Everton started to string a few passes together and their first chance of the game fell to Middlesbrough old-boy Yakubu on 13 minutes.
A long punt up field was flicked on by Louis Saha and, after out-muscling Emanuel Pogatetz, Yakubu was one-on-one with Ross Turnbull in the Boro goal, but it was the keeper who got down well to push the shot away from goal.
Everton needed something to kick start their afternoon and the Blues found it on 22 minutes after a deep cross from Joleon Lescott drifted onto the back post and rebounded into the path of Marouane Fellaini, but his waist-high volley from six yards was expertly blocked by Pogatetz.
The home side started to find momentum and in desperately searching for the equaliser they thought they had a penalty just after the half-hour mark.
Turnbull brought down Yakubu inside the box but, despite close to 30,000 claims for a spot-kick, referee Howard Webb kept his whistle in his pocket.
Everton looked most threatening from set-pieces and came close twice, both from Mikel Arteta corners, after Joseph Yobo powered a header just over the bar and Tim Cahill had the ball punched off his head from practically on the goal-line.
The Blues should have equalised just before the break after another Cahill header from a corner was drilled into the deck but the ball bounced wide of the post keeping the score 0-1 at half-time.
The Blues came out for the second half in the same spirits they left the first and instantly put the visiting backline under pressure.
It seemed only a mater of time before they would drew level and it was Yakubu who stepped up to level the scores after 65 minutes.
Good work from Arteta on the left wing won his side a free-kick deep inside the Middlesbrough half and it was he that swung the ball dangerously in towards Yakubu in front of the keeper.
After failing to connect with an attempted back header, the ball somehow managed to find its way into the net from in between his shoulder blades.
Everton were totally dominating possession and efforts from Fellaini and Yakubu had the visitors rattled, but the back four stood strong and managed to restrict Everton to long-range efforts for the last few minutes.