The 25-year-old took the captaincy in Jordan Henderson's absence and led from the front, scoring a hat-trick as the Reds secured qualification to the knockout stages of the Champions League by winning Group E.
The Merseyside club have been rampant going forward in the group stages this season, scoring 23 goals and breaking the record for the most scored by an English club at this stage of the competition.
Coutinho has been regularly linked to a January move to Barcelona but the Anfield crowd were vocal in their admiration for the Brazil international following his impressive performance and he was full of praise for the home support after the game.
"It was a really good night. When we play at Anfield the atmosphere is amazing. I always say the fans here are unbelievable. Today is a good day," said the Brazilian.
"It was a good game, the team played very well and I think we deserved the result so we are in the next round. Let's see [what we can achieve]. We have to work hard."
Liverpool welcome rivals Everton to Anfield for on Sunday, knowing that a win in the Merseyside derby could see them move into third place in the Premier League.