Matt Le Tissier has urged Southampton boss Ronald Koeman to sign striking reinforcements before the transfer window closes.
Garcia, who played for Palermo in Serie A last season but belongs to Chilean club Rangers de Talca, joins Luca Caldirola, who arrived from Inter Milan earlier this summer and plays on the other side of the back four for Bremen.
"We've been able to reach an agreement with all parties, so also with Santiago's former club Palermo," said Bremen's director of sport Thomas Eichin.
"There is nothing blocking him from playing for us now and we're delighted to have such a dynamic defender in our team."
Garcia will have a fortnight to get to know his new team mates due to the international break.
He brings with him the experience of 56 Serie A appearances with Palermo and Novara having played 14 times in Argentina's top division.