Liverpool were on top and quite dominant for about 25 minutes at the start of the game, after that Swansea took over more and more.
Liverpool started losing their grip on the game before Coutinho went off with an injury. Certainly it wouldn't have helped, but he's more of a final third player and not that involved in keeping possession deeper and playing through the first bit of pressure. Something Liverpool struggled immensely with as the game progressed against Swansea.
I think their problems were more down to their centre backs being so poor on the ball. Skrtel has always been average at best on the ball, I don't know if nerves were to blame, but Sakho looked completely lost as well. Wisdom at right back didn't help either. Without a back for that could recycle possession they were just pressed to death by an impressive looking Swansea side.
I think something similar would have happened with Coutinho on the pitch, but those few times where Liverpool did get through that initial pressure Coutinho might have been able to create something in the space that was then available.
His injury will present them with a problem, but more in terms of creating chances, not dominating games.