Chelsea will take on the French champions Paris Saint Germain while Barcelona will play Atletico Madrid.
Defender Felipe Santana and many other Dortmund players appeared offside when Santana scored the winning goal in stoppage time.
PSG's strategy for Champions League quarter-final at Barcelona will not depend on whether Messi features for the La Liga leaders, PSG coach Carlo Ancelotti said.
Real reached the Champions League semis for the third straight season after Ronaldo scored twice in a 3-2 loss to Galatasaray.
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