Goalkeeper Begovic opened the scoring for Stoke in sensational fashion after 13 seconds, a long clearance bouncing over the head of opposite number Artur Boruc to stun those in attendance at a windy Britannia Stadium.
Home captain Ryan Shawcross spurned an excellent opportunity to extend Stoke's lead in the ninth minute, but Southampton soon regained their composure and deservedly levelled through a Jay Rodriguez header in the 42nd minute.
The visitors went on to enjoy prolonged spells of possession in the second half and saw two penalty appeals turned down when Erik Pieters and Shawcross each appeared to handle in the area, the latter with one minute of normal time remaining.
After answering several questions about Begovic's goal, only the fifth by a keeper in the Premier League, Pochettino said: "Overall, I want to highlight the fact that my team played very well, even after conceding that unlucky goal.
"We showed resilience, a lot of character and identity. We were able to develop how we want to play. I think we managed the game for large phases and I want to congratulate my players, from the goalkeeper to the striker, because I think they all did really well."
Asked about Southampton's spot-kick claims, he added: "My players called for two penalties at the end of the game, but I haven't had the chance to look on TV.
"I think there's a fair chance we could have got two penalties in the second half."
Pochettino cleared Boruc of any blame for his part in Stoke's bizarre opener.
"These things can happen in football. He didn't apologise for anything because he didn't have to apologise for anything," explained the Argentine.
"As a manager, that's the first time I've had to experience that unlucky moment.
"I think it had more to do with the weather conditions. It was really hard to expect that gust of wind.
"The gusts were changing all the time, especially at the beginning of the match, and it's also a stadium that is open on three sides, which makes it even more windy.
"Overall it was just a really tough game in terms of the weather conditions."
