England survived a horror penalty count, their own errors and a furious assault from the Pumas to win 13-9 at Otago Stadium on Saturday.
It was a titanic battle between two committed forward packs and England were not assured of victory until the final whistle in a niggly, bruising encounter before a capacity crowd of almost 30,000.
Argentina could have won had they not missed six of nine penalty shots at goal, while Jonny Wilkinson kicked only two from seven penalties for England, with one conversion.
The Argentina forwards drove almost to the England line from an early lineout rumble and, from the subsequent advantage, Contepomi kicked an easy penalty to put the Pumas in front.
England equalised after 11 minutes when Wilkinson, amid ear-piercing booing from the Pumas fans, kicked a penalty from 36 metres and near touch after a high tackle by Rodrigo Roncero.
Fullback Martin Rodriguez missed two long-range shots for Argentina but the Pumas were taking it to England, with openside flanker Juan Leguizamon an expert scavenger.
Contepomi then missed from long-range, more than 45 metres on the angle, and the Pumas had missed nine points from possible goals.
England conceded its fifth kickable penalty at the end of the first quarter and this time, Rodriguez - Contepomi was receiving treatment for a rib injury - was on target from 41 metres to give the Pumas a 6-3 lead.
Argentina suffered a major blow when Contepomi was helped off after 27 minutes. He was replaced by Marcelo Bosch.
Fullback Ben Foden made the first clean break of the game for England but, from the ensuing scrum, Wilkinson unbelievably missed a sitter.
England prop Dan Cole was sin binned for pulling the scrum down after 34 minutes but Rodriguez kicked to the left from 49 metres with the resultant penalty attempt.
Argentina deserved their halftime lead but it would have been much more than 6-3 had they kicked straighter and trusted their backs more.
The Pumas started the second spell with stunning runs by Rodriguez and Bosch. Rodriguez kicked his second penalty when England infringed, taking Argentina to a 9-3 lead.
Rodriguez missed from 47 metres when England was penalised yet again but Wilkinson was not having a great night either and he missed two attempts in quick succession, one from 40 metres and the second from slightly closer.
England did try to move the ball but all their attempts were met with staunch Argentina defence.
Ledesma, who had played heroically, was subbed after 15 minutes by Agustin Creevy but two minutes later Rodriguez, from a reasonably handy position, missed his fifth penalty.
But Wilkinson recorded his fourth miss, this time from 45m and Argentina retained a six-point lead going into the last 15 minutes.
England finally took the lead for the first time 13 minutes from time. They drove from a lineout, created two rucks and replacement scrum-half Ben Youngs scooted through a gap under the posts to score.
Wilkinson converted and England were ahead 10-9. Wilkinson missed again from 40 metres, his fifth miss and no surprise by now, but he was successful from a handier position five minutes from the end, securing a hard-fought 13-9 victory.