Safiullin looked out for the count after Shapovalov made a strong start to the fourth-round tie but he weathered the storm and his steady approach paid dividends as his opponent self-destructed.
"I don't think I realised it before, but going to the third round of a Grand Slam for the first time, then to a fourth round and now to the quarter-finals, I'm super happy," Safiullin said in his on-court interview.
"I hope Denis can recover soon."
Canadian 26th seed Shapovalov raced through the first set but Safiullin hit back to break in the sixth game of the second set courtesy of a double fault, an advantage he would not relinquish.
After the match was levelled at 1-1, Shapovalov committed two more double faults in a dismal start to the third set as Safiullin broke and then held to love for a 2-0 lead.
Minutes later, Safiullin had a 4-1 double-break after yet another double fault.
The situation went from bad to worse for Shapovalov in the fourth set when he appeared to aggravate an injury to his left leg, which left him hobbling around in pain. He was quickly put out of his misery by the Russian, who broke and then closed out the victory on serve.
Safiullin becomes just the 12th man to reach the quarter-finals on his Wimbledon main-draw debut, adding to the likes of Bjorn Borg, John McEnroe and Nick Kyrgios. He next faces either Italian eighth seed Jannik Sinner or Daniel Galan of Colombia.
"It's definitely something special. I never knew these guys also were among the first ones to do this," said Safiullin, who won the Australian Open juniors title in 2015 but only broke into the ATP top 100 last year.
"It's special and I'll do my best in the quarter-finals to go as far as I can here.
"It's been hard work, but mentally I've changed since getting into the top 100. I've gotten a little more confidence in my game and I think it helps a lot."
