Everton delivered a composed performance to record their seventh away win of the season, with only Chelsea having more.
Their recent resurgence is all the more remarkable given the fact that boss David Moyes is operating without a fit senior striker.
But Cahill has stepped into the breach in the emergency and followed up his winner at Eastlands with another crucial strike to seal three Boxing Day points.
Middlesbrough, in sharp contrast, were abject and have now gone seven games without a win.
Moyes praises Everton spirit
Everton were forced to draft teenager Gosling into their line-up after Leon Osman was ruled out, along with Joseph Yobo. But it did not have an adverse effect on Moyes' injury-hit side as they dominated the first 45 minutes.
Steven Pienaar had the ball in the net early on, but it was ruled out for a push in the area as Everton pressed for an early advantage.
Tuncay then tested Everton keeper Tim Howard, but it was a rare moment of danger for the Merseysiders in the early exchanges.
Pienaar came close after 27 minutes when Cahill set him up 12 yards out, but his shot hit the post and rebounded to safety.
Middlesbrough then had a period of pressure around the half-hour mark, with Howard saving well down low to his right from Gary O'Neill and then Joleon Lescott blocking crucially from Mido.
Everton had the final chance of the first half when Lescott only succeeded in inadvertently clearing the danger after Cahill nodded Arteta's corner into the six-yard area.
The breakthrough Everton deserved came after 51 minutes at the end of a desperate goalmouth scramble.
Cahill's initial header from another Arteta corner was blocked by Middlesbrough keeper Ross Turnbull, and when another effort from Lescott was blocked, Cahill bundled home the rebound.
Middlesbrough boss Gareth Southgate needed to make a change, but when he did it was an unpopular one, with jeers ringing around the Riverside when Alves was introduced for Tuncay.
We're off the pace - Southgate
Everton should have doubled their lead after 56 minutes, but Gosling somehow turned his effort over at the far post after good work by Marouane Fellaini.
Middlesbrough put on Adam Johnson in a bid to gain some momentum, and his direct style had an immediate impact, although he picked up a booking from referee Mike Riley as he dived in an attempt to win a penalty.
Youngster Gosling had been promising for Everton, and he had another chance with 14 minutes left, only for his point-blank header to be blocked by Turnbull, who suffered a cut eye in the process.
Middlesbrough's best chance came only two minutes from time when Mido flicked on a long ball, but Alves lifted the chance over with the goal at his mercy.
Everton had held out in relative comfort, and while their fans revelled in another win on the road, Middlesbrough's supporters rounded on their players at the final whistle.
Middlesbrough manager Gareth Southgate:
"We are not full of confidence because of the run we are having. We have worked and worked but we were not good enough on the day.
"The whole afternoon was edgy and we have got a lot of hard work ahead of us. We are not in the position we want to be but there is no easy way out. It has to come from within the group.
"We are going to get a fair bit of flak and a fair bit of spotlight, but we have to come through that."
Everton manager David Moyes:
"Dan Gosling was terrific and on another day he might have got himself a couple of goals. It is a big step coming into a Premier League side pushing for Europe so he deserves a pat on the back.
"I think all the players who are playing are doing brilliantly well for us. They are great lads, but they are also good players.
"The boys are doing unbelievably with no strikers and with Tim Cahill and Marouane Fellaini working together sometimes up front and sometimes in the middle.
"They really caused a lot of problems today. It's terrific what the players are doing."
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