Already boasting a 2-0 lead from the first leg, Barca turned it into a rout as Villa and Alexis Sanchez had a brace each after Thiago's opener.
Thiago bundled the ball home in the 17th minute before a wonderful solo effort from Villa highlighted a quickfire double from the record Spain goalscorer that had Barca 3-0 up after 26 minutes. Sanchez then scored in the 55th and 85th minutes to complete the rout.
Barca finished the match with 10 men as Villa - who missed the second half of last season with a broken leg - went off with an apparent thigh problem in the 77th minute with all the substitutes already used, Sergi Roberto having also been hurt early in the second period after stand-in coach Jordi Roura made two changes at the break.
Barca now advance to face Malaga in the last eight.
They are joined in the quarter-finals by Atletico Madrid and Real Betis, although the latter only just saw off Las Palmas with Ruben Castro's 85th-minute goal giving them a 1-0 win on the night, 2-1 on aggregate.
Las Palmas had Enrique Corrales sent off in the final minute for a second bookable offence.
Getafe held Atletico to a 0-0 draw on the night but, after Rafa Lopez's 33rd-minute dismissal, had little hope of overturning a 3-0 deficit from the first leg.