Everton celebrate substitute Louis Saha's second-half winner
Everton needed a second-half fightback to beat Middlesbrough and reach the semi-finals of the FA Cup for the first time since they won the trophy in 1995.
Boro took the lead at the end of a drab first half when David Wheater nodded in Matthew Bates' cross from close range.
But Marouane Fellaini quickly levelled after the interval, beating Brad Jones to Tim Cahill's cross and heading home.
And six minutes later, substitute Louis Saha rose unmarked to nod home the winner from Steven Pienaar's cross.
Toffees boss David Moyes has not enjoyed much luck in the Cup since taking charge at Goodison Park days after they lost their last quarter-final, also against Boro, in 2002.
Everton lost to Shrewsbury, who finished bottom of League Two, in 2003 and before this season Moyes had only taken them past the fourth round once; when they lost to Manchester United in 2005.
But he has a Wembley appearance in the semi-finals against Sir Alex Ferguson's quintuple-chasers - whom Everton beat in the final 14 years ago - to look forward to after his side put on a typically tenacious display to eventually overcome an awkward Boro side.
An Everton victory did not look the most likely outcome in the early stages, which saw the visitors adjust quickest to the blustery conditions and use the fierce wind to hold up balls over the top of the home defence.
Boro had the first effort on goal after 10 minutes, with Jeremie Aliadiere firing wastefully over following a clever move down the right that saw Justin Hoyte play in Bates.
Joseph Yobo's header forced an early save from Jones as Everton came forward but the first-half quickly degenerated into a turgid midfield battle.
The closest Moyes' men came to breaking the deadlock was when Pienaar's corner flew across the face of Middlesbrough's goal and deflected inches wide off Bates.
And for long stretches, the most entertaining aspect of the tie was the wrestling match that Boro defender Robert Huth had with Fellaini whenever the ball was in their vicinity.
Fellaini only escaped Huth's clutches once; when he found space at the back post but saw his header from Phil Jagielka's cross pushed away by Jones.
Without the artistry of the injured Mikel Arteta, Everton lacked a creative spark and the home fans were voicing their frustration even before Boro took the lead a minute before the break.
Bates had time to tee up an inviting cross after picking up Yobo's clearance and Wheater rose highest to power in a header that Tim Howard could not stop from going over the line despite getting a hand to the ball.
Things looked bleak for Everton but Moyes reacted by bringing Saha off the bench at the interval and the change brought almost instant rewards.
Fellaini has now scored five goals for Everton this season
Only a timely block by Joleon Lescott stopped Aliadiere's shot after he met Tuncay's low cross at the near post.
But on 50 minutes, Everton were level when Cahill crossed from deep and, with Jones stranded, Fellaini beat Huth to loop a header into the net.
Suddenly the Boro defence appeared vulnerable, and Jones looked shaky again when he got a weak punch to a Leighton Baines free-kick, with Saha's follow-up blocked by Bates.
The home side were now looking full of confidence and, roared on by their reinvigorated support, they were soon ahead, thanks to another dangerous cross.
This time it was Pienaar who swung the ball in from the left and Saha was completely on his own as he rose to nod past Jones.
Baines almost sealed Everton's win when his whipped free-kick hit the top of the crossbar with Jones again stationary.
But Boro, the lowest scorers in the top four English divisions and missing their top scorer Afonso Alves through illness, offered little in response.
It was Everton who looked more likely to score again before the end and Saha had another effort deflected wide before firing hopelessly over when in space in the area.
Boro did belatedly threaten when Lescott had to block Tuncay's shot and Gary O'Neil's swerving free-kick almost deceived Howard but the Toffees still held out easily enough.
Everton boss David Moyes: "After a talking to at half-time, the players showed what they were really about in the second half.
"Assistant boss Steve Round did a great job on them during the break. We just needed them livened up."
Middlesbrough boss Gareth Southgate: "We've got to learn how to see matches off, especially when you get the lift of being a goal ahead.
"We knew we'd face a bombardment in the second half.
"We felt we had the players to deal with it - in actual fact for that five-minute spell we didn't.
"The Cup is always a bonus for you as a club. We had nothing to lose coming here today and had a great chance to go through. We haven't taken it."
Everton: Howard, Neville, Yobo, Jagielka, Lescott, Pienaar (Gosling 89), Rodwell (Saha 46), Osman, Baines, Fellaini, Cahill. Subs Not Used: Nash, Van der Meyde, Castillo, Jacobsen, Wallace.
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