Cahill opened the scoring and Leighton Baines made it 2-0 and despite the dismissal of Victor Anichebe and Yaya Toure's deflected goal, City could not deny David Moyes' side.
After a difficult start to the season, Cahill admitted the win was all the sweeter.
Massive: Everton had the two Tims to thank for their first win since October
'The achievement is massive for us,' he said. 'We knew we had to work hard but what's been going on in the last couple of months, we've been working, playing some good football and not got the rewards.
'But we've come to a stadium where it is hard to play and showed our mettle and played some good football too.'
Cahill was full of praise for goalkeeper Tim Howard who produced save after save for the Toffees.
In the interview with Skt Sports, he added: 'He's been the man not just this season but all the other seasons as well. It's what he brings to the team. There's times in games when we look to him to get us out of it and tonight he did that.'
Howard himself said: 'They were on top of us most of the second half and then with the sending off, we knew we had to dig in and fight, we did that to be fair to everyone.'
Moyes was similarly buoyant.
Safe hands: Cahill was full of praise the American stopper
'There's a big smile tonight, I feel terrific,' he said. 'I thought the players did me proud tonight, the way they kept at it. In the end it's a really good three points.'
While Cahill praised Howard, Moyes praised Cahill.
'He sort of takes the fight to them,' said the Scot. 'For us, Tim is the one who leads us into battle. We've thought about playing him up front, we've moved him about in different positions, we ask him to cover a lot of ground and he willingly does that.
'His attitude is terrific, and that's why he is appreciated. Everyone likes a tryer. They always like to see people who are willing to put themselves on the line for their team and he certainly does that.'
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