And the Argentinian felt his first game in charge could easily have resulted in three points rather than just the one.
Speaking through an interpreter, Pochettino said: "Tonight we have done very well and created at least seven good chances to score.
"The only shame was that we didn't score the goals but we created a lot of chances facing a very good side in Everton who are fifth in the league.
"We deserve a lot of credit although we did suffer a bit more in the second half.
"But we were creating chances and just needed that bit more luck."
Pochettino also took time to thank the Southampton fans who backed their team throughout - although they still chanted the name of sacked boss Nigel Adkins.
"I've been offered a job, an opportunity, and I've taken it," he added. "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."