The Dane cruised through the first set and was ahead 16-9 in the second, but then Popov woke up, and the French home crowd really got going in Paris.
Suddenly, Antonsen couldn't get back to his original game plan while the Frenchman grew in confidence, point by point. However, the Aarhus native finally clinched the tie when he converted his fifth match point in the match.
This means that second-seeded Anders Antonsen is ready for the quarter-finals on Friday, where 35-year-old Chou Tien-chen from Taiwan awaits.
Whether it was due to nerves in front of the home crowd remains to be seen, but Popov began the match extremely erratically.
Popov made countless unforced error allowing the Dane to take a commanding lead.
After a little more than fifteen minutes, Antonsen capitalized on his first set point and secured the set with 11 unanswered points.
Popov won the first point of the second set but the joy was short-lived. Antonsen quickly won the next three points and was back in control.
Leading 16-9, the Dane looked like he was on course for victory, but then Popov launched a terrific comeback.
The Frenchman won six points in succession, which woke up the spectators in Paris, and then the set was set to end in a close finish.
Antonsen squandered a match point, while Popov won three points in succession to force a third and decisive set.
Popov now made almost no mistakes, and this meant that Antonsen had to work much harder to get points on the scoreboard and at the break Popov was ahead 11-10.
Slowly, however, Antonsen regained his focus and began getting the upper hand in the duels thanks to strong play at the net.
Antonsen then squandered three match points but kept his cool to convert the fourth one to cut the French celebrations short.