The Argentinian's appointment came as fans' favourite Nigel Adkins was sacked and rumours were rife of a supporters' backlash towards both Pochettino and Southampton chairman Nicola Cortese.
But a scintillating start by the home side ensured the crowd stayed with the former Espanyol boss and his players as they came close to causing an upset by beating fifth-placed Everton, with Pochettino insisting he wanted focus to be on Premier League survival.
"I've been offered a job, an opportunity, and I've taken it," he said. "I'm happy how the fans have greeted me, and how the players have greeted me, and I just want to focus on my work. That's what I'm here to do.
"I have the maximum respect for the previous manager. It's never nice to take over from another manager who's been let go, but that's just how it's been."
Asked about fans' support for Adkins, he said: "That's freedom of speech. It's normal that they think that after a couple of seasons of him being in charge.
"I credit the previous manager for the work he did here. But now I really want to focus on my tenure here and the future of the club here under me, and the goals we've set out to achieve."
Southampton skipper Rickie Lambert hit the post with a free-kick and saw a header cleared off the line as the hosts created all of the chances in the first half.
Everton were much better after the interval but their boss David Moyes was left frustrated that the Toffees could not close to within a point of the Champions League places.
"I'm disappointed not to win it, but at half-time I was delighted to come in 0-0," he said.
"I saw Southampton play against Chelsea and they were really good. They're probably in their best run of form of all season.
"Some of their players are beginning to settle in and get used to the Premiership. We got what I expected in the first half.
"We've got to use every game as an opportunity to try and stay in there. We've got ourselves in a good position, but I don't know if that's bad a result: coming to Southampton and drawing. They're in really good form."