Leading 1-0 from the first leg, Suarez doubled the lead with a low finish before Lionel Messi's penalty made it 3-0 on aggregate.
Juanmi pulled a goal back before Luis Suarez made it 4-1 on aggregate.
Willian Jose's header cut the deficit again before Suarez's second goal.
Barcelona, who have won the competition a record 28 times, are through to the semi-finals for the seventh consecutive season.
They join 10-time winners Atletico Madrid, Celta Vigo, who knocked out Real Madrid, and Alaves in Friday's draw.
Spain midfielder Suarez has made more appearances from the substitutes' bench than starts in La Liga this season.
Yet the 23-year-old caught the eye with two excellent finishes that suggest he has a fine future.
His first came after Samuel Umtiti robbed Xabi Prieto deep inside the Sociedad half and from there Neymar, Messi and Luis Suarez combined to tee up his namesake for a first-time finish into the far corner.
The tie was all over by the time he scored his side's fifth of the night, the former Villarreal player skipping past two challenges before rounding the keeper and rolling the ball home.
Barcelona have won their last five games in all competitions.
Barcelona midfielder Denis Suarez: "I came with a desire to be a part of this team and I am feeling better and better.
"We started the game by dominating and once it was 1-0 they left us plenty of space.
"Everything is easier when you have the three forwards we have. We weren't nervous at all, we just wanted to dominate the game and continue on this great run that we're on."
Former Manchester City player Denis Suarez produced two wonderful finishes as holders Barcelona reached the Copa del Rey semi-finals with a 6-2 aggregate win over Real Sociedad.