Negredo capped a devastating counterattack in first-half injury time with a shot which Simon Mignolet could only palm into his own net after Vincent Kompany had canceled out Coutinho's opener.
Jordan Henderson's back heel was saved by Joe Hart and Raheem Sterling missed an open goal as Liverpool pressed for an equalizer after the break, but City held on for a big win.
City is now just a point behind Arsenal in second place and has won six of its last seven top-flight games, while Liverpool sits three points behind Arsene Wenger's side in fourth.
Martin Demichelis (knock) was ruled out, so Joleon Lescott came into the heart of the City defense. Pablo Zabaleta returned from a hamstring injury and Jesus Navas started as Gael Clichy and Edin Dzeko dropped to the bench.
Liverpool was without fullback Jon Flanagan (hamstring), so Aly Cissokho got the nod to replace him.
City wasted no time in showing its intent and was almost in front after four minutes as Aleksandar Kolarov whipped in a cross and Navas stooped at the back post to head back across goal, but was denied by the outside of the post.
Liverpool soon got into its stride and was in front after 23 minutes when Sterling got on the end of Luis Suarez's exquisite flick then rounded Joe Hart. Coutinho nipped in front of the winger to slot into the empty net from a tight angle.
City was a constant threat and it took the club only seven minutes to get back on level terms as Kompany rose to head down David Silva's corner and Joe Allen was unable to clear on the line.
Both sides were causing all sorts of problems going forward and Hart denied Coutinho a second goal after great link-up play by the lively Sterling and Suarez, before Martin Skrtel produced a vital last-ditch block to thwart Alvaro Negredo.
Negredo was not to be denied in first-half stoppage time, though, as Mignolet could only palm the striker's left-foot strike behind him and into the net after Navas had picked out his Spain teammate.
Liverpool started the second half on the front foot and Kolarov was almost made to pay for being far too casual in the area when Suarez robbed him, but the Uruguayan's shot hit Jordan Henderson and the defender's blushes were spared.
Negredo rose to meet a Silva corner but headed straight at Mignolet. Liverpool continued to ask questions of the City defense and Glen Johnson's touch let him down after Lescott's attempted clearance fell nicely for him.
Henderson forced a save out of Hart with a cheeky back heel and then Sterling spurned a golden opportunity after 72 minutes when he inexplicably fired over the bar from close range.
Suarez blasted a free kick over the crossbar from long range, but there was no way back for Liverpool.