The Uruguay internationalist joined Barcelona in the summer for £75m in the wake of another biting furore and a four-month ban, and his old strike partner Daniel Sturridge scored the winner 10 minutes from time after Nathaniel Clyne had cancelled out Raheem Sterling's first-half goal - although Brendan Rodgers still hopes to sign a striker.
"Our ambitions at Liverpool have to be bigger than one player," said the Liverpool manager in relation to Suarez. "He sent us a lovely text wishing us all the best, which was a great gesture. I texted him back. It's a real love-in. He's a great boy but he is gone. We do still want a striker but that will depend on availability."
Sturridge's goal was his 36th in 50 appearances and only George Allan, from Linlithgow Bridge, 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 Barclays Premier League's second-highest scorer with 21 after Suarez's 31 last season.
"For Daniel, it was just a continuation, showing he can get goals," said Rodgers. "It was a poacher's goal, one of those where you anticipate a knockdown. Raheem's was a great pass, great speed, touch and finish."
Liverpool finished last season with 16 wins and 53 goals from 19 games at home, a run of form Rodgers will be hoping to replicate.
"We had to grind out a result. It was a tough game and I am happy with that determination and character," he added. "You want to make a good start and that first win can set the tone - it certainly did last season.
Despite some predictions of a tough time for Southampton after a summer exodus of players and manager, they looked capable of repeating their win here last season.
"We played a great second half, after 1-1. There was one team on the pitch playing and it was Southampton," said their new manager Ronald Koeman. "We had a chance to score the second one, two great opportunities, but the quality of Liverpool . . . they don't need many chances to score and it makes the difference."