Diego Simeone's men beat Leverkusen 1-0 in the second leg to level the last 16 tie at 1-1 on aggregate before progressing to the quarter-finals on penalties.
Garcia came on in the 46th minute for Cani, but the club revealed the midfielder suffered a cracked elbow bone, sustained when Omer Toprak fell on him.
Meanwhile, striker Mario Mandzukic - who came off in the 83rd minute at the Vicente Calderon - has been diagnosed with a Grade I sprain in his right ankle following scans.
It has not been confirmed how long the respective injuries will keep the duo on the sidelines.
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