Everton's poor run of form in the Premier League continued as a single Charlie Austin strike made the difference for Southampton on Ronald Koeman's return to St Mary's on Sunday.
Koeman's men had won just one of their last seven league games heading into the encounter and the Dutchman was desperate to get back to winning ways against his former club.
The Goodison Park side remain seventh in the table following this weekend's result with 19 points from 13 games, but their hopes of finishing in the top four to qualify for Champions League football continue to fade, with fourth-placed Arsenal nine points clear.
The home side got chances to double their lead via Sims and Austin, but the former's header after a good cross from the left was straight at Everton goalkeeper Maarten Stekelenburg, while the goalscorer's low attempt after a quick attack lacked the power to worry the Dutchman, who spent last season on loan at the Saints.
Gueye should perhaps have levelled in the 26th minute after a superb team move involving Gareth Barry and Seamus Coleman, the holding midfielder wildly blasting a shot over the crossbar after the latter's cutback.
Barry then got a chance to equalise when he headed just over after a free-kick from the left, while Ross Barkley also failed to get his header on target in the closing stages of the first half after Yannick Bolasie's cross.
Austin was next to threaten for the home side with a sublime header after a cross from the right, forcing Stekelenburg into a superb save to keep his team's chances of getting a result alive.
Bolasie looked dangerous for Everton on two occasions, but the former Crystal Palace man's aim was off twice as Southampton held on for a narrow 1-0 win, with James Ward-Prowse missing two late chances to net a second.
