Cape Town - Southampton secured a deserved 1-0 victory over Everton after Charlie Austin struck inside the first minute at St Mary's on Sunday.
In a clash between two teams with indifferent form, Claude Puel's men made a dream start when Austin slotted home from close range inside the first minute from Sims' assist.
Sims could have made it 2-0 not long after but he failed to make a good connection from Nathan Redmond's excellent cross and Stekelenburg gathered.
The Merseysiders had more of the ball in the first half but were creating little although Idrissa Gueye may have done better after good work from Gareth Barry and Seamus Coleman in the 26th minute. Yet, the Senegal international fired high over the bar from the penalty spot.
After the break, Pierre-Emile Hojbjerg nearly made it 2-0 when he fired a shot in from close range, however, it hit the post even with Austin in space waiting for a cross.
The visitors had a chance to equalise when they won a free-kick just outside the box, though, Romelu Lukaku's attempt was poor and went well over in the 53rd minute.
Bolasie then went close three minutes later after being found by Aaron Lennon, but he fired just wide of Fraser Forster's post.
The action was getting frenetic and the Saints nearly doubled their lead when Austin saw his header from Cedric Soares' cross brilliantly saved low down by Stekelenburg.
The hosts then missed a sitter when Ward-Prowse skewed an effort wide from Austin's cross despite having an open goal from the six-yard box. Stekelenburg again denied the Saints youngster after he volleyed on target from the far post.