Philippe Coutinho opened the scoring on 17 minutes with a wonder strike.
The Brazil international cut in off the left wing, picked his spot from 25 yards and bent a low, curling shot past a stranded Fraser Forster.
But the Saints had an earlier penalty shout waved away by referee Roger East.
After 9 minutes, Shane Long got the wrong side of Dejan Lovren.
Long bent his run into the penalty area but then seemingly felt the defender on his back.
Long went to ground, but Lovren appealed that the contact was not enough to justify the foul.
The referee agreed and play continued.
After going one behind, Ronald Koeman’s men chased an immediate response but go caught with a sucker punch on 21 minutes.
Saints were caught short at the back as Divock Origi played the ball to Daniel Sturridge who bamboozled an isolated Ryan Bertrand 12 yards out.
The striker dipped a shoulder to throw Bertrand off the scent and buried a low shot into the far corner through the space left behind.