Begovic netted a freak goal after just 13 seconds when his long kick caught the wind and then bounced over the head of Saints keeper Boruc.
Jay Rodriguez later equalised to secure a point for Southampton, who remain in the top five after a strong start to the season.
Begovic admitted after the final whistle that he felt sorry for Boruc, insisting there was never a moment when he thought the ball would go in.
He told Sky Sports: "It was just one of those things. I got fortunate with the good bounce and obviously the pitch was still a bit wet. They were tricky conditions with the wind and it went win.
"I feel bad for Artur Boruc. It is never nice to be on the receiving end of one of those but we needed a goal and it helped us get a point in the end.
"I just tried to get it forward. There was a bit of pressure on me and it just kept carrying and carrying through the wind and it is just one of those fortunate things. It went for us this time."
Boruc offered no excuses for conceding the goal and apologised to his team-mates, but stressed that he was determined to make up for the mistake.
"This is my job and I should protect those kind of situations and in the end I didn't do anything," said Boruc.
"That cost us two points and it is hard for the lads because they put a lot of effort into today's game and probably deserved three points.
"What can I say? I can promise I will get those two points back."