The Saints got off to a flying start and top-scorer Rickie Lambert squandered numerous opportunities in the opening 45.
After the break, the game slowed down and neither side could find a winner.
Below, Sports Mole picks the bones out of an unmemorable first match for Pochettino.
On target 11
On target 7
Was the result fair?
Not really. Southampton were brilliant apart from at least a goal in the opening 45, but faded in the second. Everton had chances of their own and could have easily nicked it.
Blisteringly good in the first half but lacking drive in the second. How Lambert didn't score in the first half will be puzzling many. Twice he was superbly denied by Howard before missing a free header by inches. He also hit the post with a curling free kick and clipped the bar after the interval. Defensively, Southampton were more solid than usual and keeper Artur Boruc did what he needed to when called upon.
Perhaps one of their worst of their season. Moyes's men really didn't show up on the south coast and are lucky to be heading back to Merseyside with a point. They improved marginally after half time, but this is mainly because the opposition faded out.
Sports Mole's man of the match
Luke Shaw: The Saints left-back is only 17 but he put in a mature performance full of pace, decisions and technique.
Only one winner here, Everton forward Nikica Jelavic.
Maybe the easiest game of Neil Swarbrick's time in the Premier League. There were no contentious decisions to be made and the match ran very smoothly.
Southampton: Pochettino's task doesn't get any easier as the Saints next go to league leaders Manchester United in eight days' time.
Everton: The Toffees are in the fourth round of the FA Cup though and go to Bolton Wanderers of the Championship on Saturday.