O'Neill was announced as Stoke's new manager last week, earning a 4-2 win over Barnsley in his first game.
But he will remain in charge of the Green and White Army until the end of their Euro 2020 qualifying campaign, whether that ends in the next seven days or extends into March.
Northern Ireland face Holland in Belfast on Saturday and Germany in Frankfurt on Tuesday in their final Group C fixtures, knowing automatic qualification is highly unlikely but they are currently well placed for the play-offs.
For many in the squad, including Smith, O'Neill is the only international manager they have known as he has led Northern Ireland since 2012, but aside from a short discussion he said it has been business as usual for the team this week.
"We had a brief meeting and he just said 'Thank you' and said without the players here he wouldn't have had the opportunity to go and take the job at Stoke," defender Smith said.
"Then it was basically straight to business with two tough games coming, and we need to win them."
The Hearts right-back said there was no surprise that O'Neill was finally heading into the club game, having been linked with several vacancies since guiding Northern Ireland to Euro 2016.
"You look at the job he's done with limited resources, we've really probably overachieved in the last few years but we're still doing it," he said.
"It was only a matter of time before he took one of those jobs. The timing's probably right."
It might have happened sooner, but Smith believes it was never going to be an easy move for the 50-year-old.
"It will be hard for him to leave this," Smith said. "He's built something special here and for him to leave it and take another job is probably a massive decision for him.
"He's felt it's the right time and everyone supports him in that. He's done all he can here and no one can say anything bad about him."
Automatic qualification for Euro 2020 is not yet impossible for Northern Ireland if highly unlikely, and Smith said the focus was on winning Saturday's meeting with Ronald Koeman's side at Windsor Park to keep those faint chances alive.
"It has to be, otherwise we're not going to come out in these two games and perform well," he said.
"We haven't spoken about the play-offs this week at all. It's solely on Holland first, and then we'll turn our attention to Germany."
The Holland game will be of double importance not only to keep those slim automatic hopes alive but also as it likely to be O'Neill's final home game in charge, subject to a change in the play-off permutations.
Northern Ireland can take encouragement from their performance in Rotterdam, where Holland needed two stoppage-time goals to complete a come-from-behind 3-1 win last month, and would dearly love to send O'Neill off in style.
"He's had an incredible eight years," Smith said. "He says probably the one thing missing from his CV here is a really big scalp, so that would be fantastic to send him off from Windsor that way."