The Andalusian outfit held a 6-1 advantage from the first leg but Sabadell took the lead through a 22nd-minute penalty from Juan Collantes.
That was as good as it for the Segunda Division side, though, as Kevin Gameiro equalised 10 minutes before the break and Aspas scored in the 59th, 60th and 62nd minutes to blow the visitors away before Gerard Deulofeu made it 11-2 on aggregate nine minutes from time.
Barcleona have one foot in the last-16 after a 4-0 win over third-tier Huesca in the first-leg of their fourth-round tie.
Ivan Rakitic opened the scoring after 12 minutes and Andres Iniesta doubled the lead four minutes later before Pedro Rodriguez made it 3-0 six minutes before the break.
Rafinha then completed the rout 18 minutes from time to give Barca a comfortable cushion to take to the Nou Camp.
Atletico Madrid also claimed a comfortable win over third-tier opposition as they beat Hospitalet 3-0.
Antoine Griezmann finally broke the deadlock after 67 minutes and a Gabi penalty and a Cristian Rodriguez strike in the last 10 minutes sealed the win.
Deportivo La Coruna and Malaga drew 1-1 in their all-Primera Division contest as Jose Toche’s strike cancelled out Nacho Camacho’s opener.
In the night’s other first-leg tie, Jhon Cordoba’s 25th-minute goal gave Granada a 1-0 win over Malaga, who saw Patrick Ekeng sent off 19 minutes from time.