Raheem Sterling hit the post in an otherwise tame first-half showing from either team, but Martin Skrtel felled Zach Clough just before the hour mark in breathe life into the game.
Eidur Gudjohnsen duly powered home only for Neil Danns's second yellow card to set up a nervy finish.
Jordan Henderson hit the post as the game entered the final 15 minutes and, with just three minutes left on the clock, Raheem Sterling volleyed in a dramatic equaliser.
With the game entering extra-time, Coutinho curled in an injury-time winner from the edge of the area as Liverpool prevented an embarassing exit.
