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The Blaugrana have been in impressive form since their 3-2 loss to Athletic Bilbao in the Copa del Rey on 20 January, which they avenged with a 4-0 league win last time out.
Ahead of the meeting between two of their rivals in the race for the Champions League places, Real Betis and Atletico Madrid, Barca looked poised to have to settle for a point.
Elche took the lead in the 44th minute through Fidel, but Ferran Torres ended his much-publicised wait for a league goal on the hour before Depay blasted home emphatically from 12 yards after the VAR adjudged Antonio Barragan to have handled in the area.