An equaliser for Germany seemed certain in a match that Vidic's side desperately needed to win, but when Lukas Podolski drilled the spot-kick towards the bottom right-hand corner, goalkeeper Vladimir Stojkovic saved it to spare Nemanja further agony.
"I was so relieved," Vidic told FIFA.com following his team's 1-0 victory.
"It was my mistake. I just got disorientated - for a moment I thought there was someone behind me - and I handled the ball (from Podolski's cross).
"Fortunately 'Stojk' produced a fantastic save. I said to him afterwards, 'I owe you a gift. What do you want?'
"But thinking about it now, I should be asking him for a gift. Look at the attention he's getting – he’s a hero. I gave him that opportunity to become a hero!"
From starting first-round elimination in the face, Serbia are now guaranteed to reach the last 16 if they beat Australia in their final Group D match in Nelspruit on Wednesday evening. A clash between the top two sides, Ghana and Germany, will kick off in Johannesburg simultaneously, at 20:30 local time.