The Italian, playing his first tournament since losing to Novak Djokovic in the Australian Open semi-finals last month, eased to a 6-3, 7-5 second-round win in Doha.
Sinner will play Jakub Mensik in Thursday's quarter-finals.
The world number two has now won his last 50 matches against players ranked outside the top 50, a run dating back to a loss to Dusan Lajovic in Cincinnati in 2023.
"I respect every player, but I always try to play my best tennis," Sinner said when told of that streak.
"The ranking, at the end of the day, is just a number. Everyone is playing (with) high quality.
"Especially when the opponents don't have much to lose, you always have to stay very focused."
Australian world number 53 Popyrin battled gamely but failed to create a break-point opportunity against his clinical opponent.
Sinner dropped just three points on serve in an excellent first set, which he took courtesy of a break in the sixth game.
Popyrin fought hard in the second but could not force a tie-break as Sinner broke to grab a 6-5 lead before confidently serving out the set.
World number one Alcaraz takes on Frenchman Valentin Royer in his second-round match later.