First-half penalties from Iniesta and Fabregas were added to by Sanchez in the 71st minute and Pedro a minute from time at Camp Nou, while the closest the visitors came was when Piti thundered a shot off the crossbar with the score still at 1-0.
Barcelona made light of being without the injured duo of Lionel Messi and Victor Valdes, Iniesta opening the scoring in the 20th minute following a foul on Fabregas, before the roles were reversed 20 minutes later.
In the second half, Neymar played in Sanchez and Fabregas teed up Pedro to put the seal on another comfortable win for Gerardo Martino's side.
Barca's victory was their fifth in succession and their 13th in 14 overall in La Liga, while the setback for Granada came as a blow after winning three of their past four matches.
Barcelona made six changes from the 4-1 win over Real Betis a fortnight ago with Messi (hamstring) and Valdes (calf) among a host of absentees for Martino, while Granada showed three alterations from their 3-1 win over Malaga.
The hosts went close in the 14th minute as Roberto reacted smartly to parry Neymar's free-kick from the left.
But Barca did not have to wait long for the breakthrough as six minutes later Fabregas was brought down in the box by Dimitri Foulquier and Iniesta stepped up to plant the subsequent spot-kick into the bottom right-hand corner.
Granada almost responded in spectacular fashion four minutes later, Piti driving forward and unleashing a thunderous effort from 25 yards that cannoned off the crossbar and behind to safety.
Jose Pinto, called in as replacement for Valdes, held Piti's 25-yard free-kick on the half-hour mark, before Barca doubled their lead five minutes before the break, again from the penalty spot.
Fran Rico brought down Iniesta as he prepared to shoot and this time Fabregas took responsibility, burying his spot-kick high into the top left-hand corner.
Five minutes into the second half, Pedro wasted a glorious chance to put the result beyond doubt, blazing the ball over an open goal from 10 yards after Roberto had palmed a low cross into his path.
Allan Nyom should have halved the deficit moments before Barca's third, drilling the ball straight at Pinto from close range before the hosts broke at speed, Neymar sliding a pass for Sanchez to finish with a minimum of fuss.
In the final minute, Fabregas broke away down the left and squared for Pedro, who slid in to convert from eight yards and cap another impressive Barcelona performance.
