The Springboks needed a Handre Pollard penalty after England collapsed a scrum three minutes from time to win the match 16-15 in a thrilling finish to a match that Borthwick's side looked set to win.
South Africa will now play New Zealand in the final, back at the Stade de France next Saturday, while England will play Argentina in the third-place play-off at the same venue 24 hours earlier.
"The whole team performance was strong," said Borthwick. "We're disappointed, we came here to win the game and we fell a bit short.
"They played their hearts out. It was an intense encounter, we're disappointed we came out on the wrong side of it."
Borthwick was full of praise for the Springboks, saying "they are the world champions for a reason".
"Immense credit to South Africa to come back and find a way to win. Great credit to them, they are the number one side in the world," he said.
"They found the way to gain the advantage around the scrum and that gave them the field position and the points."
Borthwick argued, however, that his team had a bright future ahead of them.
"If you look at this squad it is a very good squad of players. We had seven players 25 or under today, that’s the most of any semi-finalist. South Africa had one," he said.
"This is a really talented group from the experienced to the not so experienced."
Captain Owen Farrell, who kicked four first-half penalties and a monster drop-goal for all of England's points as they raced 15-6 ahead, agreed, saying: "We have made a start of going forward and there is a lot to come from this group."
Farrell said the last few months had been "a rollercoaster" for the England team, who came into the World Cup on the back of a historic first defeat by Tier II Fiji at Twickenham.
"I'm glad where we have built to, but gutted we don't have a crack at the big one next week. I am massively proud of this group," he said.
"We came up with a plan during the week and the weather conditions played a part in it as well. We started the game really well, we shocked them at times and they made a few changes to change what they were doing.
"But credit to them fighting their way back into it and finding a way to win at the end."
Borthwick said that attention would now be fully turned to the third-place play-off against Argentina.
"We'll look forward to it. We wanted to be in the final on Saturday, now we play Argentina back here on Friday and we'll prepare thoroughly for it," he said.
"It will be a week when we step forward again."
