Nathan Redmond's first-half strike was enough to secure a 1-0 victory for Claude Puel's side in Wednesday's (11 January) first-leg clash at St Mary's where the home team should have perhaps scored more.
Without Sadio Mane, who is currently on international duty with Senegal at the Africa Cup of Nations, Liverpool toiled on the south coast and it was only after the introduction of Coutinho after 61 minutes did they begin to pose any sort of threat.
It was the Brazil international's first appearance since late November, having suffered an ankle injury in a Premier League match against Sunderland that would rule him out for the next six and a half weeks. His return to action is a timely one however, with a trip to Manchester United looming large on Sunday (15 January) and Klopp insisted post-match the playmaker came through his return match with no problems.
"No, no, all good," Klopp told Sky Sports when asked if there had been any reaction to Coutinho's recent injury. "It is good to have him back and you could see immediately in the few moments how important he is for any team in the world, for us especially, so it's all good."
Liverpool skipper Jordan Henderson also missed the trip down south due to a heel problem that has ruled him out of the club's opening games of 2017, but is expected to return to training on Thursday.
Joel Matip was also absent despite returning to training after an ankle injury earlier this week.
Manchester United can close the gap on arch-rivals Liverpool to just two points with victory at Old Trafford on Sunday. Klopp's side however have lost just twice in the league all season.