Cahill was given a golden opportunity to open the scoring before Baines doubled the lead 15 minutes later with an exquisite finish to take the sting out of City's tail.
The result sees David Moyes' men continue their fine run on their travels this term, but having had the lion's share of possession, Mancini's side will ultimately be left to rue the missed chance to leapfrog their cross-city rivals and top the table, just hours after Carlos Tevez withdrew his transfer request.
It took the visitors only four minutes to make their mark at Eastlands, courtesy of some uncharacteristically lax defending from City. Seamus Coleman got on the end of a probing Baines cross and when he put the ball back into the box, Cahill was left unattended in the six-yard box to gratefully nod home the simplest of finishes.
A City response was quick in the making, though, and when a swift counter-attack ended with Yaya Toure curling a delightful pass over the Everton back four, you would have expected David Silva to do better than bring the ball down and wastefully rifle it into the side netting.
With City continuing to push forward, there was a sense of foreboding inevitability concerning an equaliser. However, against the run of play, the marauding Baines again charged down the left flank and, after overlapping Victor Anichebe, the full-back received a first time lay-off from Cahill inside the box and bent the ball, with his weaker right foot, around the despairing Joe Hart and just inside the far post.
From then onwards, Mancini's shell shocked side appeared to have their work cut out for them, but continued to take the game to Everton . After a lengthy spell of possession, Pablo Zabaleta found himself free in the box and his low driven cross nearly made its way through to Carlos Tevez, only for Jack Rodwell to boot the ball out for a corner, which was subsequently wasted.
A flurry of activity in and around the Everton box around the half-hour mark resulted in a number of City efforts on goal, most of which were blocked by the visitors who resorted to putting large numbers behind the ball in order to stifle the waves of City attacks.
Despite all the pre-match fanfare for Tevez, the diminutive Argentina striker remained largely anonymous throughout the opening exchanges, his only contribution being to draw a foul from Marouane Fellaini, which resulted in the Belgian receiving a booking for dangerous play.
From the resulting free-kick, a ball into the Everton box was caught by a hopeful flick from Tevez which saw the ball loop high up into the air and almost drop underneath the relatively untroubled Tim Howard's bar, only for the American to catch the ball and stumble out of play to concede a corner.
With City continuing to dominate possession, Howard was drawn into action again shortly afterwards, when a viciously bending Aleksandar Kolorov free kick from thirty-yards out forced the goalkeeper to dive full stretch to his left hand side and palm the ball away.
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Seeking to add more cutting edge to his team in the final third, Mancini brought on Adam Johnson as a second-half sub at the expense of James Milner. The England winger made an immediate impact, swinging some handy balls into the Toffees box, but to no avail.
The game took a nasty edge when Fellaini was perhaps lucky to not be sent for an early bath, with another studs-up challenge, this time on Mario Balotelli, escaping referee Peter Walton's attention.
City continued after the interval in much the same vein they ended the first half. Lengthy spells in possession were not translated into clear-cut chances, with two rasping Kolorov drives from just outside the area representing the hosts best efforts on goal, the first drawing a smart save from Tim Howard, the second dragging just wide.
With the hosts dictating the game, the task of reducing the deficit appeared to become a little easier when Anichebe was given his marching orders after being shown successive yellow cards for needless fouls, leaving his team to face the last 30 minutes a man down.
Sustained bouts of City pressure came to little fruition, with David Moyes' side, who were heavily camped inside their own half, managing to frustrate the hosts by crowding them out any time they got the ball into the final third.
The staunchly resilient Everton defense was finally breached in the 72nd minute when Adam Johnson found himself free in the box, slipping the ball to Yaya Toure whose shot took a wicked deflection off the otherwise imperial Phil Jagielka and spun into the far corner of the goal.
The hosts were then within inches of securing the equaliser five minutes on. A ball over the top was brought down on the chest of Balotelli and when the Italian poked the ball beyond Howard, it trickled agonisingly towards goal and bounced off the post. The rebound was met by both Howard and Tevez and the 'keeper made a fantastic save to prevent the City captain from sinking it into the net.
As they constantly drove forwards City looked increasingly likely to find the back of the net once more but found an unassailable wall in the form of Howard. First the American beat away a beautifully hit drive by Yaya Toure before denying Tevez with an even more impressive close-range save.
The closing minutes of the game matched the pattern of the rest of the encounter with City failing to make the most of their utter domination of possession and with a late red card being shown to Kolo Toure, Everton managed to desperately hold on to climb to 14th in the table.