The Bosnia international's wind-assisted long punt down field took a huge first bounce over Southampton counterpart Artur Boruc after just 13 seconds.
However, despite being handed such a fortuitous start, the Potters were pegged back by Jay Rodriguez's header just before half-time as the match finished 1-1.
"I definitely meant it," joked Begovic after becoming only the fifth goalkeeper to score in the Premier League after Peter Schmeichel, Paul Robinson, Brad Friedel and Tim Howard.
"It was nice. I've never scored a goal so I'll take it. I've never come close as I've never been allowed up for corners but it happened to go in.
"I didn't think it was going in at all and when it went in I thought 'What?'.
"It was a little bit surreal but the ball got caught in the wind and carried a distance.
"I'm in pretty good company so I'll take it as it helped us get a point.
"It was a shame we couldn't carry it on and get a second goal and really get on the front foot.
"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."
Begovic did, however, joke he was happy to pass on scoring tips to any team-mates struggling to find the net.
He told BBC Radio 5 Live on Sunday: "There wasn't so much banter (from my team-mates). Anyway, 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."
Begovic also had some sympathy for Boruc, who was on his penalty spot when the ball was launched from inside Stoke's area but suddenly found himself in trouble when centre-backs Dejan Lovren and Morgan Schneiderlin allowed it to bounce just outside their own box.
"It is a terrible feeling to be on the end of one of those but that's football," he told Stoke Player.
"One bad bounce for a goalkeeper makes you look silly."
It is not the first time Boruc has been beaten from distance but he admits the previous occasion was in a youth game.
"The last time it happened I was a kid and it was on pitches without the stands and a bad surface," said the Pole after conceding only his fourth goal in 10 league matches this season.
"I knew it was going to be bad conditions so it's a big thing in the warm-up to do it properly. I was very angry to concede.
"To be honest I didn't have enough time to judge the ball especially after the bounce, I knew that I was in the middle of nowhere.
"The thing is, that's a goalkeeper's life and sometimes that situation happens and you need to bounce back as soon as possible.
"For the next game in a week's time, I'll take time to prepare properly myself.
"I promise to everybody that I will get those two points back."
Having recovered their composure after the early shock of conceding such a freak goal Southampton gradually took a hold on the game and deservedly equalised when Rodriguez headed in a cross from James Ward-Prowse.
They had a much better spell when they really applied the pressure in the opening 25 minutes of the second half but they could not get themselves ahead when they were on top.
"To be honest we deserved a little bit more than the one point," Boruc told Saints Player as the side made it seven matches unbeaten.
"Hopefully we're going to keep this great run of performances going in the next couple of games and throughout the season as well."
Southampton manager Mauricio Pochettino praised his side's ability to bounce back from adversity.
"I want to highlight the fact that my team played very well even after conceding that unlucky goal," he said.
"It showed resilience, character and identity. I think we managed the game during large phases and I want to congratulate my players from the goalkeeper through to the striker."