The opening goal of the game came after nearly 80 minutes when Brazilian defender David Luiz put Chelsea in front while Ramires completed the scoring in injury time to hand Carlo Ancelotti's side all three points.
The victory moves the Blues up into third, above City, and Essien told Sky Sports that the club will fight for every point until the end of the season.
"We have to believe that we can do it and there is a long way to go and we’ll fight for every point," he said.
"It was a very tight game, we take every game as it comes and we’re still working really hard to win games. So we’ll see what happens."
David Luiz has now scored two important goals since his arrival from Benfica and Essien hailed the performances of the centre-back at the start of his Blues career.
"Another goal from him today and since he came in he’s been fantastic," he added. "We need these type of players at Chelsea and he’s come in and done a great job."