The Potters took the lead in bizarre fashion after just 13 seconds when Begovic's clearance bounced over Artur Boruc and into the net.
However, Jay Rodriguez struck back in the 41st minute, and Begovic admits that his side had to dig deep to earn a point.
"It was a shame we couldn't carry it on and get a second goal and really get on the front foot," the Bosnian told BBC Radio 5Live.
"But Southampton have been on a great run and are full of confidence so we knew it would be tough.
"We had to graft out a result but we'll take the point and take the positives out the game."
When asked about his goal, the 26-year-old joked that he is now giving goalscoring advice to his teammates.
He said: "I definitely meant it. I'm joint top-scorer now so it's working pretty well in my favour at the moment.
"If they want any pointers, they know where I am and I'll pass it on to them."
Next up for Stoke is Swansea City on Saturday.