Middlesbrough's poor home form continued as Andrew Johnson and James McFadden struck in the second half to give Everton a battling win.
Johnson, who had a hand in both goals, celebrates his opener
Johnson latched on to Steven Pienaar's flick to steer Everton ahead with a precise low finish past Mark Schwarzer.
And McFadden combined superbly with Johnson, showing great composure and neat footwork to make it 2-0.
Boro wasted several good chances with Tuncay going close and Gary O'Neil forcing a fine save from Tim Howard.
Boss Gareth Southgate will once again be ruing Middlesbrough's inability to take the limited chances on offer.
They certainly had the better of the opening hour with the appearance of Yakubu in Everton blue rather than Boro red at the Riverside sparking the home crowd into life in a frenetic opening.
Yakubu's every touch was booed and both sets of players appeared equally pumped up and ready to get the season of goodwill out of their system.
With the rain cascading down and surface water on the pitch, crunching challenges flew in from both sides but opportunities were at a premium throughout.
Everton sorely missed the creativity of Mikel Arteta and the energy and goalscoring threat of Tim Cahill in midfield.
The visitors had only lost two of their last 16 games, but they offered little going forward in the first half.
McFadden, deputising for Arteta, went closest but his low, long-range shot dribbled just wide.
Boro, clearly buoyed by Saturday's 1-0 win away at Portsmouth, played with a confidence and inventiveness hardly befitting a team who had won just one home game in their last six attempts.
The lively Tuncay should have put Boro ahead when he was sent clear inside 10 minutes but a heavy first touch allowed Tim Howard to dive bravely at the striker's feet and the ball was cleared.
And a fine move almost resulted in Stewart Downing side-footing Gareth's Southgate ahead.
Things will turn for us but we need to get a bit of belief back and get the place bouncing again.
Fabio Rochemback and Luke Young combined on the right and the latter's cleverly dinked cross found Downing but Phil Jagielka was in the right place to deflect the ball off the line.
Boro continued to look the more likely of the two sides to score after the break
Downing did well on the left and fizzed a cross across goal, which struck Tuncay and bounced harmlessly to Howard.
Tuncay then raced clear and, when his shot was only half-cleared, Howard reacted quickly to pull off a fantastic save, denying O'Neil from close range.
Yakubu blazed wildly over the bar at the other end after a clever back-heel from Pienaar.
Just as Boro started to run out of ideas, Everton seized control.
Johnson showed his quality in front of goal to put them ahead and McFadden sealed the points when he exchanged passes with Johnson before guiding the ball past Schwarzer to take Everton up to fifth spot.
Middlesbrough manager Gareth Southgate:
"As is often the case with games at the end of the Christmas period, the first goal has a massive influence.
"If we'd have got it, and we had two cleared off the line, it would have been a huge lift, an extra boost. But they got it and the energy drained away.
"We weren't quite at our best but I can't fault what they've given. There wasn't the freshness to win the game. They're out on their feet and they must be feeling like every time they take a step forward a door slams in their face.
"There's a thin line in this league. Things will turn for us but we need to get a bit of belief back and get the place bouncing again. At home we have found it a bit tough."
Everton boss David Moyes:
"We had some good bits of fortune at the right moments but I have to praise the players.
"We put on a great show and I feel we deserved it. It was stalemate for a long time, they had moments but we played well enough to get the points.
"We've no Tim Cahill, no Joseph Yobo or Leon Osman and Thomas Gravesen is injured. We came here with just about no midfield. Our only recognised midfield players were Lee Carsley and Steven Pienaar.
"It shows the strength in depth and what we have got here. The players who came in were fantastic."
Middlesbrough: Schwarzer, Young, Woodgate, Huth, Pogatetz, O'Neil (Johnson 73), Rochemback (Cattermole 84), Arca, Downing, Sanli, Lee (Hutchinson 59).
Subs Not Used: Turnbull, Riggott.
Everton: Howard, Hibbert, Jagielka, Lescott, Nuno Valente, McFadden (Baines 80), Carsley, Neville, Pienaar (Anichebe 90), Johnson, Yakubu (Vaughan 87).
Subs Not Used: Wessels, Stubbs.
Goals: Johnson 67, McFadden 72.
Ref: Mike Riley (Yorkshire).
BBC Sport Player Rater man of the match: Everton's James McFadden 6.63 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.