There was a more fruitful outcome for another Australian contender, Christopher O'Connell. The qualifier, ranked No. 82, edged out Lebanese wildcard Hady Habib 7-6 (9-7) 7-6 (7-3).
And the reward for O'Connell is an immediate match-up with Draper in the next round in Doha.
Draper was back in action for the first time in over a month, since he retired when two sets down to Carlos Alcaraz at the Australian Open.
The 23-year-old had previously been troubled by a hip problem in the weeks leading up to Melbourne.
But Draper looked in decent shape as he broke big-hitting Popyrin twice when racing to the first set in just 27 minutes.
The second set was a tighter affair, with Draper having to save a set point at 5-6.
But he stood up well in the tie-break, finishing the job with a pair of aces to come through a good test of his fitness. Now O'Connell awaits.
Elsewhere, the top-seeded Alcaraz overcame Croatian veteran Marin Cilic 6-4 6-4 to also reach the round of 16.
Cilic, 36, the 2014 US Open champion who has fallen to No. 192 in the rankings, was playing his first match of the season after recovering from a long-term knee injury.
The Croatian led 4-3 and 0-40 on Alcaraz's serve in the second set, but the Spanish four-time major winner managed to recover and then broke in the next game.
"I am just really happy that I stayed calm at that moment, breathing, going through my routines, making good points," Alcaraz said. "That's why I was able to win in two sets."
Alcaraz will now play either Zhang Zhizhen of China or Italy's Luca Nardi.
Several other top players will play their first matches later on Tuesday.
Australia's second-seeded Alex de Minaur will face Roman Safiullin. Novak Djokovic, who had to pull out of his Australian Open semi-final with a hamstring injury, will take on Matteo Berrettini.
Daniil Medvedev, Andrey Rublev and Stefano Tsitsipas also play on Tuesday.
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