The Senegal international forward spoke out following the Reds' 2-2 opening Champions League Group E clash at Anfield on Wednesday evening when Coutinho stepped off the bench on 76 minutes.
It was the Brazil international's first appearance of the season a month after handing in a transfer request in an attempt to force a move to Barcelona.
Judging by the reaction of Anfield to his introduction Coutinho has been forgiven by fans, as they and Sane is also looking forward to his full-time return.
"It is always nice because he is one of the best players so we welcome him back and he is an important player for us to get better and better - not only my game but I think he is a very important player for Liverpool," said the 25-year-old.
"We always need him so we are happy that so we wish him a welcome back and he will help us to get better and better."
Meanwhile, Mane has also spoken out about the red card he received for kicking Manchester City goalkeeper Ederson in last weekend's 5-0 defeat at the Etihad Stadium and insisted that he only intended to win the ball.
He added: "My intention was not to hurt the goalkeeper. It was bad and I wish him the recovery to come back healthy.
"I think it was a challenge for the ball. Honestly, I was going for the ball, I didn't have intention to hurt him because I am not that kind of player so I wish him a quick recovery.
"I made contact and it was not what I wanted to do so I apologised to him. It is part of football so I cannot change anything.
"I don't have his number so I just posted messages for him. I think he saw them."