It was the third successive home win for the Argentine side and their fourth in five matches overall in their second season in the competition.
The match, the first in Super Rugby between Argentine and Australian teams, was scoreless midway through the first half when Reds wing Eto Nabuli and lock Kane Douglas were yellow carded within a minute of each other.
The home side opted for a scrum when awarded a penalty close to the Reds line, and number eight Leonardo Senatore picked up and passed to centre Jeronimo de la Fuente, who darted over with Hernandez adding the conversion.
Seven minutes later, wing Ramiro Moyano scored a fine try, his fifth in three matches.
Fellow wing Bautista Ezcurra made the initial thrust in a move from inside the home side's half before passing to Moyano, who weaved past several Reds to dive over, though flyhalf Juan Martin Hernandez missed the conversion.
An exchange of penalties by rival flyhalves Jake McIntyre and Hernandez as halftime approached saw the teams change ends with the Jaguares 15-3 ahead.
The Jaguares added a third try, De la Fuente's second, early in the second half.
The forwards pushed the Reds pack back 20 metres in a rolling maul and with the line in sight, the centre ran on to a pass and darted through a gap to dive over.
The final half hour belonged to the Reds in possession and territory after number eight Scott Higginbotham scored their try when he went over in a maul.
But the home defence kept them at bay, even when they lost Ezcurra to a yellow card from the 63rd minute.
The win put the Jaguares second in the Africa 2 Conference, one point behind the Lions, while the Reds remain third in the Australian table.
(Editing by Peter Rutherford)
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