Southampton were much the better side in the first half as in-form striker Rickie Lambert hit the woodwork with a bending free-kick and narrowly failed to score with two trademark headers.
Everton improved in the second half but also failed to convert chances as the home side clung on for a valuable draw.
Everton moved to 38 points, within three of fourth-placed Tottenham Hotspur and four ahead of Arsenal and Liverpool while Southampton edged further away from the relegation zone to 23 points, four above the drop.
Former Argentina international Pochettino was controversially appointed manager despite Southampton's improved recent form, including a 2-2 comeback draw from two goals down at European champions Chelsea on Wednesday, Adkins' final game in charge.
Supporters of the south-coast club had threatened a frosty reception for the former Espanyol boss, given Adkins' popularity and the fact he had lifted Southampton out of the relegation places after leading them back to the Premier League.
"I want to thank the players for the three days they have worked with me, they have done well. I also have to thank the supporters, they have been kind and the reception was amazing," Pochettino told Sky Sports through an interpreter.
"We did well tonight, we created several chances to score. The only shame was that we did not score a lot of goals but we created a lot of chances against a good side like Everton. We suffered a bit in the second half, we were not as good as we were in the first half.
"I really enjoyed the game today. The people who have worked in this league always said that it is amazing."
There was little sign of hostility towards Pochettino or executive chairman Nicola Cortese as Southampton began brightly and Everton defender Phil Jagielka nearly turned the ball into his own net early on.
Lambert crashed a 30-metre free-kick on to the post from 30 metres and Everton goalkeeper Tim Howard was called into action on several occasions to deny the buzzing hosts.
Two Lambert headers, a Gaston Ramirez effort and defender Jos Hooiveld's turn and shot underlined Southampton's dominance of the first half.
Home goalkeeper Artur Boruc was forced into his first genuine save after 51 minutes when Marouane Fellaini shot low at the far post only for the Pole to clear with his leg.
From the resulting corner Fellaini headed straight at the Southampton keeper and Everton started to find their feet, Nikica Jelavic spurning a great chance when he failed to connect with Victor Anichebe's inviting low cross.
Boruc made a great one-handed save high to his right from Anichebe after 68 minutes when the substitute was played through and fired off a fierce left-foot shot.
"I was frustrated with how we played and frustrated in the second half that we did not get the goal," Everton manager David Moyes said.
"Southampton have played well in recent weeks and they continued that form. We got better in the second half. In the end they had the better of the first half and we probably had the better of the second."
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