The former Burnley striker came to Liverpool with high hopes three season ago, but struggled with injury and lack of first team football, scoring just four goals in 25 appearances for the Reds.
His move to Southampton went through in the final few minutes of the transfer window on Thursday evening.
Jurgen Klopp paid tribute to him during Friday's press conference ahead of Sunday's Premier League opener with West Ham.
“It is a very strange feeling saying goodbye to Danny," said the Liverpool manager. "The biggest emotion is how much we will miss him.
“He is such an outstanding boy. Honestly, if you could bottle character and spirit, his would sell worldwide. He is so incredibly positive and full of energy. But we won’t just miss his character because he is a footballer who has all the tools.
“What is clear is that in his time here he had zero luck really, certainly in my time. Before joining I tried to watch as many games on tape as I could from the season so far and Danny Ings stood out as someone I was really looking forward to coaching.
“We all know his story and the bad luck he has had since then. It’s why the other emotion is that of being pleased for Danny that he has such a good move and his future is now very bright.
“When he said to us he thinks this is the window it might be best to look elsewhere, for the sake of his career and playing more regular football, it was clear early on that Southampton has great appeal to him personally. It’s his home, his family is there, it’s his boyhood club. They have a great manager and I think the supporters and him will feed off each other’s passion and energy.
“So we say farewell to a friend, but for Danny I know the best chapters in his personal story are still to be written. He is as fit now as at any point of his career I would bet and he just requires the opportunity.
“He has that opportunity and on behalf of his Liverpool family we say good luck Danny and we will miss you.”