Southampton outplayed the visitors for much of the game, but their finishing let them down during a bright early spell. Rickie Lambert had the best chances, but came closest when his free kick from 25 yards rattled the post.
Everton improved after the break but was held scoreless for the first time in 10 away games.
Pochettino was greeted with a warm reception by Southampton fans after taking over following the firing of Nigel Adkins.
''I think today we've done very well, we've created at least seven good chances to score,'' Pochettino said through an interpreter. ''The only shame was that we didn't score any goals. ... We just needed that bit of more luck.''
Southampton is now four points out of the relegation zone in 15th place, while Everton remains in fifth, three points behind fourth-place Tottenham.
Lambert caused havoc for the Everton defense in the first half but lacked sharpness in his finishing. He had a header cleared off the line by a combination of Nikica Jelavic and Baines, and then headed wide from 12 yards after a pinpoint cross from Jason Puncheon.
''It was a good point but it's disappointing not to get all three,'' Lambert said. ''Especially in the first half we played some really good football and I should have put at least one, maybe two away.''
Everton then nearly grabbed the opening goal after the break, with Marouane Fellaini twice being denied by Artur Boruc in quick succession.
''After the way we played in the first half I was frustrated,'' Everton manager David Moyes said. ''In the second half we got good opportunities but we couldn't nick a goal.''