The boot of Owen Farrell proved the difference as both sides scored two tries but the England captain was unerringly accurate as he put over four penalties and two conversions for a tally of 16 points.
Sam Underhill, a late call up to the squad and making his tournament debut, delivered a man-of-the-match performance, making 24 tackles, as he reprised the form that made him one of the star players in England’s run to the final four years ago.
"It is a bit surreal. It's been a great few weeks, I've loved being part of this group," he said.
"We just stuck to our game plan, we didn't panic. A lot of the lads know it is the last 20 minutes of these games that matter. We stuck to our plan and that is testament to them."
Friday's clash was, of course, the match neither team wanted to be playing in but both coaches said this week that they were giving it the utmost respect and the physicality and commitment from the start showed that the players felt the same way.
Six weeks ago on the opening weekend England recovered from having flanker Tom Curry sent off in the third minute to deliver a totally dominant display in a 27-10 victory over the Pumas.
This time, also after three minutes, Curry, winning his 50th cap after a difficult week with his involvement in accusations of discriminatory language against South Africa hooker Bongi Mbonambi, stuck his bandaged head and strapped nose straight into a ruck to win the first penalty.
Against a barrage of boos from the largely French crowd, Farrell duly slotted it and the scene was set.
England then scored an excellent try as Farrell and Marcus Smith flicked quick passes for a charging Ben Earl to crash over. Farrell converted and added another penalty for a 13-0 lead.
Argentina’s attack was brought up short by too many handling errors but they did eventually string together a sharp move, albeit with what looked a clear forward pass, that finished with scrumhalf Tomas Cubelli squeezing through for a try.
Emiliano Boffelli’s conversion and an earlier penalty meant it was 16-10 at the break.
EXPLOSIVE START
There was an explosive start to the second half as the ever-elusive Santiago Carreras slipped through three England tackles to score under the posts.
The lead lasted less than two minutes, however, as hooker Theo Dan, who was the worst culprit for the try, got immediate revenge by charging down Carreras and picking up to score.
A Boffelli penalty closed the deficit to three and there was then a scruffy 15-minute spell as replacements poured on, kicks went astray and tackles were missed.
It was England, though, turning the screw and it paid off when they earned a scrum penalty that Farrell slotted, only for replacement Nicolas Sanchez to do the same for the Pumas to set up a tense finale.
Having been overhauled by South Africa so heartbreakingly three minutes from the end of their semi-final last week on the same pitch, England found themselves in a similar position as Argentina won a penalty with five minutes left.
However, from far out on the left, Sanchez pulled it wide.
This time England saw out the final minutes and, though the prize was not of remotely the same magnitude, it still meant they finished the tournament with six wins from seven games and bronze medals to show for it.
"You have to be able to play rugby a few ways and we are developing that and hopefully it will keep improving," said Farrell after England made it 12 wins in the teams' last 13 meetings.
"We want to show this team fight for every minute of the game. In a scrappy game like that, we proved that again."
England coach Steve Borthwick added: "It wasn't a game of incredible high quality but one of high tension. Immense credit to Argentina for the way they played and the way they have gone through this tournament.
"You can see both teams have progressed through it and while it wasn't a classic of free-flowing rugby, it was a tight affair."
Michael Cheika, whose future as Argentina coach remains uncertain, was unhappy with the officiating.
"We didn't get the rub of the green on many things but we still stayed in the game," he said.
"I'm disappointed with the way the game was refereed and the consistency. I feel for the lads as they deserved more than what they got out of today."
It was the first time England had taken the bronze medal at the World Cup after winning the 2003 tournament and finishing runners-up on three other occasions.
