With Chelsea playing in the League Cup final later in the day, City could have moved within two points of the leaders, but Coutinho's stunning strike instead ensured Liverpool claimed a crucial victory in their bid for a top-four finish.
Jordan Henderson brought the game to life after 11 minutes with his third league goal of the season for the hosts, the 24-year-old finding the top corner from 20 yards.
Manuel Pellegrini's side pushed for a second equaliser, but to no avail and City remain five points behind Chelsea having played a game more, while Liverpool are up to fifth.
A frenetic start was dominated by Liverpool – who made four changes from their defeat to Besiktas on Thursday – with Adam Lallana twice going close.
Lallana's first effort was saved by Joe Hart after a superb pass from Coutinho and, although the second beat the City goalkeeper, it was ruled out for offside.
Coutinho was again involved as he robbed Vincent Kompany before combining with Raheem Sterling to get the ball to Henderson, the Liverpool captain giving Hart no chance with a powerful shot into the top-right corner.
City – who also made four changes following midweek European disappointment against Barcelona – responded strongly, Aguero sending a low drive against the post a minute after Henderson's opener.
Lallana had a chance to restore Liverpool's lead before the break from a chipped Lazar Markovic pass, but pulled his effort wide of Hart's goal.
Both sides wasted chances early in the second half as Aguero and Sterling failed to convert from inside the penalty area, while Lallana had a second effort ruled out by the assistant referee.
Silva then spurned a promising position for City with an over-hit free-kick.
