Pochettino replaced Nigel Adkins as Saints manager last week in a move that angered a large majority of Southampton fans.
But, despite rumoured protests aimed at chairman Nicola Cortese, the supporters gave the former Espanyol boss a warm reception and his team lit up St Mary's as they outplayed the Toffees for large periods of the game.
The result moved Southampton four points clear of the relegation zone and Pochettino was quick to thank everyone involved at the club for the impressive start to his career on the south coast.
"They have shown me a great reception," he said.
"(It was a) great effort in a complex situation, and taking into account we've only had three days to prepare, I think we've done very well.
"I just want to say thank you to the players for their efforts, and to the fans for making that effort to be here, and for all the kindness they've shown to the team."
The Argentinian said he could not let the shadow of Adkins' sacking linger and did not mind hearing the home fans cheer for their former manager during the 90 minutes.
"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.
"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."
Despite seeing his side forced into their own half for almost the entire opening 45 minutes, Everton boss David Moyes believes leaving St Mary's with a draw was a missed opportunity to move within one point of the top four.
"We've got to use every game as an opportunity to try and stay in there," he said. "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."
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 came back at them after the break and had a number of chances to snatch all three points, leaving Moyes deflated at having to settle for just one.
"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 Premier League. We got what I expected in the first half."