Los Blancos ran out 4-0 winners in the first leg at the end of October, and therefore their progression was a foregone conclusion ahead of the second leg in Madrid.
Melilla had two attempts inside the first three minutes of the match, with 18-year-old attacker Mohamed Mizzian heading wide of the post before Yacine Qasmi struck over the crossbar from outside the box.
The Segunda B outfit were full of confidence at the Bernabeu as they continued to force the issue early on, and Madrid goalkeeper Keylor Navas had to keep out a low strike from Mizzian in the 13th minute.
The home side had a wonderful chance of their own moments later when Asensio found Isco inside the box, but the Spaniard could not find a route past Melilla goalkeeper Pedro Moreno.
The lively Mizzian then struck just over the Madrid crossbar once again as chances continued to arrive.
Melilla goalkeeper Moreno was called into action in the 20th minute to keep out a strike from Vinicus, before Mariano Diaz was denied a simple tap-in by a wonderful Mohamed Mahanan challenge in the 28th minute.
Vinicius somehow fired over the away side's crossbar in the 32nd minute during a strong spell for Los Blancos, and the breakthrough finally arrived one minute later.
Asensio has struggled for consistency this season, but the Spaniard beat a number of players before driving into the penalty box and finding the bottom corner with an excellent finish.
Asensio's first arrived in the 33rd minute, and he had his second less than two minutes later as he smashed a low Vinicius cross into the back of the net.
Melilla had been so bright in the first 30 minutes, but folded late in the half, with Real scoring their third in the 39th minute when Sanchez turned an excellent Asensio pass over Moreno and into the back of the net.
Isco made it 4-0 early in the second period when he found the top corner with a stunning hit from outside the box.
Federico Valverde struck wide of the Melilla post just past the hour, before Moreno kept out Vinicius once again as Real looked for more goals against their third-tier opponents.
The home side had their fifth in the 75th minute when Vinicius drove into the Melilla box before firing home at the second time of asking, having initially been denied by Moreno.
Melilla managed to get their name on the scoresheet in the 81st minute, however, when Qasmi smashed past Navas from the penalty spot after the former Paris Saint-Germain forward had been fouled by Sanchez.
The scoring was not finished though, with Isco registering for 6-1 in the 83rd minute to make it seven goals at the Bernabeu on a mild Thursday afternoon.
REAL MADRID (4-3-3): Navas; Carvajal (Garcia 46'), Vallejo, Sanchez, Odriozola; Llorente (Ceballos 46'), Valverde, Isco; Asensio, Mariano, Vinicius (Fidalgo 78')
MELILLA (4-4-2): Luis; Romero, Mahanan, Richi, Jilmar; Ruano (Brahim 56'), Ortega, Menudo (Martin 60'), Oscar Garcia; Yacine, Mizzian (Otegui 46')