Asensio’s double came in quick succession amid a flurry of activity late in the first half as he found the back of the net twice before defender Javi Sanchez added a third inside seven minutes.
Isco made it 4-0 two minutes after the break and then, after 18-year-old Vinicius Junior had got his name on the scoresheet, doubled his tally.
Melilla striker Yacine Qasmi left the Bernabeu Stadium with something to celebrate after converting a late penalty, but the Segunda Division B, Group 4 leaders bowed out on the wrong end of a 10-1 aggregate scoreline as Santiago Solari’s men handed him a seventh win in eight outings as head coach.
Having left the newly-crowned Ballon D’Or winner Luka Modric out of the 18, Solari had to be patient and it was not until the 33rd minute that the LaLiga giants got their noses in front when Asensio ran on to full-back Dani Carvajal’s ball and fired into the bottom corner of Pedro Luis’ goal.
Once the seal had been broken, Real had the bit between their teeth and they extended their lead within two minutes when Asensio repeated the feat with teenager Vinicius Junior this time supplying the ammunition.
The game, as well as the tie, was effectively over six minutes before the break when Asensio turned provider to allow Sanchez to help himself to a third, and Melilla’s hopes of keeping matters respectable were deal a further blow within two minutes of the restart when skipper Isco blasted home from distance.
Vinicius went close twice in quick succession and then had two more efforts saved as the home side went for the kill, but he finally got his reward with 15 minutes remaining when he rounded off a pacy break.
Qasmi made no mistake from 12 yards after being fouled by Sanchez as time ran down, but the respite was temporary and Isco restored the five-goal margin seven minutes from time.