The Jaguares and Blues battled away in persistent rain in Auckland which detracted from a spectacle which was not helped by 24 penalties, 15 of them by the Blues.
The Jaguares scored first with an Agustin Creevy try while the Blues were down to 14 men with Akira Ioane in the sin bin.
The Blues replied with tries to Tumua Manu and Matt Duffie, with a penalty by Stephen Perofeta to lead 13-5 at the end of an extended first half.
It continued for an extra seven minutes as the Jaguares took repeated scrums from five consecutive penalties on the Blues line but they were unable to score and the referee saw no need for another yellow card nor a penalty try.
The Argentinians stepped up the pressure as they demolished the Blues pack in the second half and were rewarded with tries to Emiliano Boffelli and Tomas Lezana with a conversion and penalty by Nicolas Sanchez.
The win moved the Jaguares into the top eight to the delight of captain Joaquin Tuculet.
"We are very happy," he said.
"In the second half we played very well and we won the game. The forwards, their work was amazing and the scrum and lineout, so our win is for them."