French midfielder Geoffrey Kondogbia headed them in front from a corner after 32 minutes and Spain striker Alvaro Negredo headed the second from an Ivan Rakitic cross in the 70th minute.
Granada conceded a third when Chile midfielder Gary Medel was left unmarked at the back post and volleyed in following a corner five minutes later.
Sevilla, who travel to Atletico in the Cup on Thursday, climbed to 11th in La Liga with 26 points from 21 games. Granada stayed 17th with 20 points.
Barcelona, who thrashed Osasuna 5-1 on Sunday, top La Liga with 58 points, 11 ahead of second-placed Atletico who lost 3-0 at Athletic Bilbao.
Barca are 15 points in front of third-placed Real Madrid who swept Getafe aside 4-0.
Holders Barca visit Real in the other Cup semi-final first leg on Wednesday.