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.
"I'm happy, I want to help the team whenever I play", he admitted.
He said in his post-match press conference: "There's no time to think about it - we are not even sure who is in".
The fixture was a curtain raiser for Sunday's Copa Libertadores second leg final between Argentinian rivals Boca Juniors and River Plate.
Asensio struck twice in three minutes and Sanchez added a third in a flurry of activity just before the break.
Midfielder Isco then extended the advantage three minutes into the second half, netting his first since September a year ago, while Vinicius Jr. scored his maiden goal for the Spanish giants as Madrid strolled into the next round easily. According to a report from OK Diario, the reigning European champions would only allow the former Malaga star to leave for a whopping €700 million.
Real Madrid players celebrate after entering the last-16.
Asensio's contribution will have done him little harm after starting the previous two games on the bench, and Solari is delighted to have competition for places.
Nacho is out for Real Madrid, though they should be able to start a full-strength backline.