Sadio Mane's 31st-minute header and a late Thiago Alcantara strike – his first for the club – proved the difference as the hosts ensured their slim hopes of clinching UEFA Champions League qualification remain alive.
The Saints, meanwhile, failed to make themselves mathematically safe from relegation as they suffered defeat, but with 10 points separating them from 18th-placed Fulham, it is highly unlikely they will be dragged into the mix.
Liverpool's last two outings – 1-1 draws against Leeds United and Newcastle United – were largely defined by a failure to take goal-scoring opportunities, rather than any issues creating them.
Jurgen Klopp could have been forgiven for worrying that a familiar pattern was once again emerging as his team missed chance after chance in the opening 25 minutes.
Mohamed Salah was guilty of twice finding Fraser Forster's gloves when he might have done better, while Diogo Jota and Mane each shot over from good positions.
When Alisson did brilliantly to close the angle on a Che Adams one-on-one created by Nathan Tella just after Georginio Wijnaldum hit the bar, it looked like the Reds were in for another forgettable outing.
However, from the resulting attack, Liverpool went ahead, Salah sending in an in-swinging cross that Mane simply could not fail to head home from close range.
The fragility of that lead was underlined before the break when Tella headed straight at Alisson with the goal at his mercy, and the Brazilian needed to show sharp reactions again to deny substitute Michael Obafemi just past the hour mark as the Saints pushed.
Clearly enjoying his starring role, the goalkeeper even gave himself another huge stop to make as he passed the ball to the feet of Adams before getting down to block the subsequent shot.
A nervous finish could have been avoided had Mane stayed onside before lashing home a recycled ball beyond Forster late in the second half, but the hosts wrapped things up in the 90th minute through Thiago's low finish from the edge of the box.