England were in total control in Marseille as they led 24-10 past the hour-mark, but two quick tries from the Pacific Islanders levelled the scores and swung the momentum in the latter's favour.
The cool head of captain Owen Farrell put England back on top with a drop-goal and a penalty in a 30-24 win that set up a semi-final against hosts France or South Africa in Paris.
After coming into the tournament with five defeats in six games, it has been a quick turnaround in fortunes for Borthwick.
"These (England) players were written off," he said. "A lot of people said we wouldn’t get out of the pool. We got out of the pool, now we got out of the quarter-final.
"I am sure we will be written off again, but these players rise to the occasion."
Borthwick always believed his team would be ready for the World Cup despite the difficult build-up.
"I don’t care what other people think about us, I only care about the team. I am privileged to work with a fantastic group of players," he said.
"I said the team would be ready and they were. They have grown through the tournament. Not so long ago we might not have come back to win this game, but this team did. I see a composure in them."
Borthwick preferred to give credit to Fiji rather than dwell on how his side let their commanding 14-point advantage slip in just a few minutes.
"For large parts of the game we controlled it," he said. "There was a spell when Fiji scored back-to-back ties which they can do well, probably better than anyone else in the world.
"But we stayed composed and got back ahead on the scoreboard."
Pressed on who he would prefer to face in the semi-finals, Borthwick would not be drawn: "I have no control over that game. We will watch it with a lot of interest."
Farrell kicked 20 points and believes the result is a big step forward having come through a tight contest.
"It was what we expected. They are a tough, tough team that can turn it on in the blink of an eye," he said. "We always knew Fiji were going to have some good patches.
"To find a way to win and get through to the semi-finals is a big step forward but we know we have plenty of work to do."
Farrell said England would relish the semi-final with renewed belief.
"We will prepare the way we have been over the past six weeks, we will enjoy going into a big week and we will attack it (the challenge)," he said.
Flyhalf rival George Ford kicked three drop-goals in England’s pool stage win over Argentina, and Farrell was pleased to have landed another with the scores tied at 24-24.
"The forwards did a great job and put it on the plate for us," he said. "I just had to knock it over."