The Pumas paid the price for allowing England to build a 13-0 lead and while they fought back to control the second half, Borthwick's men held their nerve to send scrumhalf Ben Youngs into Test retirement with a bronze medal.
Their only defeat at France 2023 was the agonising 16-15 loss to South Africa in the semi-final.
"I'm delighted for the players to get the win after they have worked so hard," Borthwick said.
"We won six games out of seven and lost one game by one point to the current world champions and current world number one, which shows the progress of the team. It shows how the team has built during the tournament.
"Playing finals games at World Cups is important. In the last two World Cups this group of players have played six finals games and won four of them.
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