Koeman, who left Saints in the summer, could have no complaints about the result as his side produced a lack-lustre performance and were second-best for long periods.
Everton started the match appallingly and Southampton took the lead inside the first minute when Charlie Austin nodded the ball over the line from a yard out after it had bounced off Josh Sims.
Saints almost doubled their lead on the quarter-hour mark when the lively Sims headed straight at Maarten Stekelenburg in the Everton goal when the youngster should have done better.
After Austin had gone close to grabbing his second goal of the match in the 25th minute, Everton finally created their first opening but Idrissa Gueye wasted the chance after he had been set up by Seamus Coleman.
Veteran midfielder Gareth Barry headed a presentable chance over the bar and, as Everton began to get on top towards the end of the first half, Ross Barkley headed Yannick Bolasie’s cross wide.
Claude Puel’s team started the second period well and Pierre-Emile Hojbjerg went close to making it 2-0 before Stekelenburg saved Sims’ shot. When Everton tried to hit back, Romelu Lukaku’s poor free-kick flew over Fraser Forster’s goal.
Stekelenburg made an outstanding stop just before the hour mark to palm away Austin’s header and James Ward-Prowse stabbed an effort wide before the Everton keeper somehow kept out another Ward-Prowse shot.
Everton struggled to create many second-half openings, although Coleman did have one goalbound effort blocked and, but for Stekelenburg, Koeman’s side would have suffered a heavier defeat.
Pedro stepped off the bench to send Chelsea into the FA Cup semi-finals after they beat Leicester 2-1 at the King Power Stadium on Sunday evening.
RB Leipzig came from behind at the Red Bull Arena to end a run of four winless games and defeat Bundesliga leaders Bayern Munich 2-1 on Sunday.