Ronald Koeman went for three changes from Sunday's win over Athletic Bilbao, with Sergi Roberto, Sergio Busquets and Francisco Trincao restored to the line-up.
Granada launched the first opening on 20 minutes through a smart delivery from Roberto Soldado from good hold-up play, but Antonio Puertas directed wide.
However, the host waited just 13 minutes to go ahead after poor defending from Samuel Umtiti allowed Alberto Soro to gain possession and set up Kenedy to tap home.
Barcelona had it all to do as the second half commenced, whilst Granada introduced a double change in Dimitri Foulquier and Darwin Machis who replaced Kenedy.
The Blaugrana were stunned yet again when Soldado brought a ball from Angel Montoro in his stride two minutes in before going on to beat Marc-Andre Ter Stegen with a calm finish.
Trincao began to question goalkeeper Aaron Escandell with a cracking strike from a promising area, but he rattled the woodwork.
Barcelona maintained the pressure late on as Ousmane Dembele cracked the crossbar from a well-worked move from close-range.
The match looked to go down the wire on 88 minutes when Antoine Griezmann's speculative strike hit the post before ricocheting off Escandell and into the net.
It did indeed take that route as Jordi Alba popped up in an advanced territory to profit from close-range following a ball from Griezmann, as the match went to extra-time.
Barcelona went ahead ten minutes in extra-time through a powering header from Griezmann after meeting a cross from Alba.
It lasted only a few minutes as Granada levelled from the spot courtesy of Fede Vico following a foul on Carlos Neva by Sergio Dest.
The Catalans regained the lead in the 108th minute after Escandell parried Lionel Messi's shot straight to Frenkie De Jong, who needed no second invitation.
It got worse for the host three minutes later when Alba connected with a ball from Griezmann to beat Escandell to see his side into the semi-finals