With their league position causing much cause for concern Gareth Southgate would no doubt have arrived on Merseyside seeing Boro's FA Cup run as a welcome distraction.
With Brazilian striker Jo cup-tied and Tony Hibbert, Mikel Arteta, Victor Anichebe, James Vaughan, Nuno Valente and Yakubu sharing a treatment room, putting a decent team together for any game is becoming increasingly difficult for David Moyes.
Boro midfielder Mohamed Shawky missed the game after picking up a knee injury during a training session on Friday, but Robert Huth was restored to the side, while misfiring Afonso Alves did not even make the bench. Like Everton striker Jo, his Boro counterpart Marlon King was cup-tied.
Boro started the game with real purpose, taking the game to Everton from the off. The Toffees were struggling to string a pass together as Tuncay, Gary O'Neil and Stewart Downing kept carving the home side's defence open.
In the 11th minute Matthew Bates set up Tuncay with a lovely cross but the Turkey international shot marginally over Tim Howard's bar from 15 yards. Everton were rattled and in the early stages were reduced to sending nothing balls in makeshift striker Tim Cahill's general direction.
The Boro fans sensed an upset and the 5,000 or so who made the trip from Teesside easily outsung the home faithful.
It was the Toffees though who could have taken the lead in the 30th minute after a Steven Pienaar corner sailed tantalisingly close to the goal, but evaded everyone to roll harmlessly out.
A minute before half-time Boro got the goal their play deserved when David Wheater rose above Joseph Yobo to head powerfully at goal from three yards.
Howard got a hand to it, but the ball spun over the goal-line and though the USA keeper swept it back into play, there was no doubt the ball had crossed the line.
Everton returned for the second half with Louis Saha replacing youngster Jack Rodwell, as Moyes looked to shake things up after his side's insipid first-half showing.
It took Everton six minutes to get back on level terms after Cahill delivered a high cross into the Boro area where the giant frame of Fellaini hovered and looped the ball over Brad Jones' despairing hands into the Boro net.
Everton were a changed side and on 56 minutes were also ahead. After a spell of pressure on the Boro goal, Leon Osman crossed hard across the box for Pienaar to chase across the pitch, the diminutive South African delivered a wicked cross close to the Boro near post and sub Saha glanced a beautiful header over the keeper.
Leighton Baines could have made it three with a great free-kick after Cahill was scythed down 20 yards out. The former Wigan man stepped up and rasped a sweet drive which looked headed for the net. The home crowd groaned though as the ball bounced harmlessly off the top of the crossbar and into the air.
After a tepid first half the game was eventually looking like a cup tie with challenges flying in as some passion was at last being displayed.
It was becoming end-to-end stuff, though Everton were beginning to look the part and stretching Boro's defence, with Saha and Fellaini linking really together.
Boro were not rolling over though with Tuncay almost levelling with a shot that Joleon Lescott just managed to get his body in front of.
Middlesbrough went searching for the equaliser with Everton happy to sit back and attack on the counter.
As the game entered injury time, just to keep the home fans' hearts in their mouths, Howard spilled an O'Neil free-kick but luckily for him Lescott cleared the danger and ended Boro's involvement for another year.