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Barca crashed out of the Champions League with a 3-0 defeat at Bayern Munich midweek, but it looked set to respond with a battling victory at Estadio El Sadar.
Nico put Xavi's side in front, only for David Garcia to equalise a couple of minutes later.
Ez Abde appeared to have won it with a strike early in the second half, but Avila struck in the 86th minute to rescue the home side.
