First-half penalties from Andres Iniesta and Cesc Fabregas put the Catalans in command at the break.
Granada were then left to play the final quarter of the game with 10 men as Manuel Iturra was shown a second yellow card.
Alexis Sanchez added a third soon after from Neymar’s pass before Pedro Rodriguez rounded off the scoring a minute from time.
Barca’s title rivals Real Madrid and Atletico Madrid can close the gap later on Saturday as they face Almeria and Getafe respectively.
With Victor Valdes, Xavi Hernandez, Dani Alves and Jordi Alba also out injured, Barca boss Gerardo Martino was forced into making seven changes from the side that started their last outing against Real Betis two weeks ago.
However, Martino reiterated his belief that Barca have an ample enough squad to cope with such a number of key absentees.
“Some of the best players, like Leo or Victor, are missing, but all the players are very good. Those that can play just now are very good so I don’t see it as a different match. It was a game in which we had some difficulties but there was no motive for us not to play a good game.
“We have to work out how to win every game we are faced with and the best thing we can do is wait for Leo, Victor, Jordi, Dani, Xavi whilst winning so that the team gains confidence and so they don’t have to come back in a week or a month with the extra pressure of begin the saviours of the team.
“The thing that I will never do is make excuses.”
The opener arrived after 20 minutes when Dimitri Foulquier brought down Fabregas and captain for the day Iniesta took responsibility to slot the spot-kick low into the corner of the net.
Granada came close to an equaliser four minutes later with their best effort of the game as Piti drilled a shot from 25 yards off the crossbar.
However, it didn’t take long for Barca to double their advantage as Fran Rico clipped Iniesta’s heels just inside the area and this time Fabregas stepped up to rifle the ball into the roof of he net.
It should have been three for Barca moments later when Pedro somehow blazed over when presented with an open goal after Roberto had flapped at Martin Montoya’s cross.
The visitors’ task became even tougher 23 minutes from time when Iturra saw his second yellow card for upending Pedro.
Sanchez made it 3-0 when he lifted the ball over the advancing Roberto from Neymar’s well-weighted through ball five minutes later.
With the game won 17-year-old Adama Traore replaced Neymar to make his Barca debut and he nearly sealed it with a memorable goal as he weaved past three Granada defenders before Roberto smothered his low effort.
And Pedro put the shine on another comfortable afternoon for the Spanish champions when he turned home Fabregas’ cross.
Real Sociedad moved up to sixth thanks to four goals from Mexican striker Carlos Vela as they came from 3-1 down to beat 10-man Celta Vigo 4-3 at Anoeta.
Two goals from on-loan Barcelona midfielder Rafinha Alcantara had put the visitors in charge, but Vela scored twice in three minutes after Andreu Fontas had been sent-off to complete a remarkable return to form having not previously scored in La Liga since August.