Tim Cahill pounced at the third time of asking to send Everton on their way to their seventh Barclays Premier League away win of the season and pile more woe upon Middlesbrough boss Gareth Southgate.
Cahill needed three attempts to force home the scrappiest of winners in the 50th minute but it was a well-deserved victory for David Moyes' men against a Boro side facing a bleak new year.
The home side have now not won at the Riverside Stadium for almost two months and the displeasure was clear with a chorus of boos accompanying Southgate up the tunnel at the final whistle.
Despite Gary O'Neil bringing a fine save out of Tim Howard and a strong second-half penalty appeals involving substitute Adam Johnson, the visitors were always in control and should have had a second goal through debutant Dan Gosling.
This despite Moyes being forced to once again start a game without a recognized striker and the absence of both Leon Osman and Joseph Yobo increasing their already lengthy injury list.
Gosling looked bright as the visitors dominated the opening exchanges, with Steven Pienaar seeing a 10th-minute effort ruled out for a foul in the box by Joleon Lescott.
The visitors were enjoying most of the early possession with Marouane Fellaini causing plenty of problems in his rangy role, but Boro had the first shot on target when Tuncay hit a low 14th-minute shot straight at Howard.
Julio Arca's awful back-header almost gifted the visitors a 27th-minute lead as Cahill raced to pick up the loose ball and set up Pienaar who blazed a rising shot against the top of Turnbull's right-hand post.
Then another mistake by Arca allowed Phil Neville to launch a rare run on goal ended with a low, long-range effort which flashed inches wide of the diving Turnbull's right-hand post.
Boro went even closer in the 33rd minute when Downing's cross from the right fell to O'Neil on the edge of the box and his driven right-footed shot was superbly turned around the post by Howard.
With the home side enjoying an all-too-rare period of control, Downing's ball into the box was picked up by Mido whose shot on the turn was blocked at close range by the alert Lescott.
Gosling had a shot blocked in the box by Pogatetz but with referee Riley frequently stopping the action for minor infringements it was developing into a niggly and forgettable affair.
Cahill broke the deadlock in the 50th minute with a goal befitting such a scrappy tie, poking home at the third attempt from a move started by Mikel Arteta's right-wing corner.
Things should have become worse for the home side after 57 minutes when Fellaini lobbed a cross from the right which found Gosling unmarked on the edge of the six-yard box but the youngster blazed high and wide of target.
Substitute Johnson almost made an immediate impression when he ran into the box into the 71st minute and fell under the challenge of Arteta, but referee Riley waved away furious penalty appeals.
Boro toiled in their bid to get back into the match with the visitors coping well with what little threat they could muster and frequently offering more danger on the counter-attack.
And it was left to Alves to wrap up another woeful afternoon for the home side when he chipped a clear chance over the bar in the 88th minute after being played into the box by Mido.