The Spaniard restored City's lead with the help of the Belgian keeper's error after Vincent Kompany had cancelled out Coutinho's opener in a thrilling clash at the Etihad StadiumAlvaro Negredo's seventh goal in nine home matches saw Manchester City defeat Liverpool to move into second in the Premier League.
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 cancelled out Coutinho's opener.
Jordan Henderson's backheel was saved by Joe Hart and Raheem Sterling missed an open goal as Liverpool pressed for an equaliser after the break, but City held on for a big win.
VIEW FROM THE ETIHAD STADIUM
By Tom Maston
On another day Liverpool may have left Manchester with at least a point, but when you fail to take your chances and your goalkeeper makes an error on the cusp of half-time, then you can have no complaints.
Brendan Rodgers may point to Raheem Sterling’s wrongly ruled out goal, but in truth his team had enough chances to equalise in the second half.
For Manuel Pellegrini and City, the hard work starts now. They sit second having played the rest of the top eight at home, and though their 100 percent record at the Etihad Stadium may remain until the season’s end, their indifferent away form must improve if they are to beat the big boys on their travels.
Martin Demichelis (knock) was ruled out, so Joleon Lescott came into the heart of the City defence. Pablo Zabaleta returned from a hamstring injury and Jesus Navas started as Gael Clichy and Edin Dzeko dropped to the bench.
Liverpool were without full-back Jon Flanagan (hamstring), so Aly Cissokho got the nod to replace him.
City wasted no time in showing their intent and were 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 their stride and were in front after 23 minutes when Sterling got on the end of Luis Suarez's exquisite flick then rounded Joe Hart and Coutinho nipped in front of the winger to slot into the empty net from a tight angle.
City were a constant threat and it took them 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 team-mate.
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 defence 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 backheel 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.