The controversial Uruguay international, who joined Barcelona in the summer for £75million in the wake of another biting furore and four-month ban, sent his well wishes to manager Brendan Rodgers and his former team-mates on Sunday morning.
His old partner Daniel Sturridge did his best to ensure Suarez will not be missed with the winner 10 minutes from time after Nathaniel Clyne had cancelled out Raheem Sterling's first-half opener - although Rodgers is still in the market for a top-quality striker.
"Our ambitions here at Liverpool have to be bigger than one player," said the Reds boss in relation to Suarez.
"He sent us a lovely text this morning wishing us all the best, which was a great gesture. He is a friend now of Liverpool.
"I texted him back. It's a real love in! He's a great boy but he is gone now.
"We do still want another striker but that will depend on the availability. Today was about winning and we got there."
Sturridge's goal was his 36th in 50 appearances and only George Allan, with 41 in the late 19th century, has scored more goals in his first half century of games for Liverpool.
The England forward was the Premier League's second-highest scorer with 21 after Suarez's 31 last season and he will be expected to contribute even more this campaign.
"For Daniel, it was just a continuation, showing he can get goals," added Rodgers.
"I think that is 36 goals in 50 appearances which is (almost) the best of any striker in Liverpool's history. That shows the level he is at.
"He always looks like he will score. It was a poacher's goal, one of those where you anticipate a knockdown.
"And Raheem's was a great goal, a great pass, great speed, touch and finish."
Towards the end of last season Liverpool were blowing away opponents at Anfield and finished the campaign with 16 wins and 53 goals in 19 games at home.
This performance did not come close to that but it was never really expected after a summer disjointed by the World Cup and the arrival of new signings.
"Most club's pre-season would have been broken with players coming back late in terms of fitness and organisation," said Rodgers.
"We had to grind out a result. It was a tough game and I am really happy with that determination and character.
"You know you want to make a good start and that first win can set the tone for you - it certainly did last season.
"As the season rolls on, we will get better and more fluent."
Despite the critics predicting a tough time for Southampton after a summer exodus of players and manager they performed well and during the second half looked capable of repeating their victory here last season.
"I think we played a great second half, after 1-1 I think there was one team on the pitch playing and it was Southampton," said new boss Ronald Koeman.
"We had a chance to score the second one, two great opportunities but the quality of Liverpool, they don't need many opportunities to score and that makes the difference.
"The way we played today I think brings us a lot of confidence."
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- Brendan Rodgers
- Raheem Sterling