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 want to thank the supporters for the very kind reception they have given me, it was amazing," he added.